| L(s) = 1 | − 6·3-s + 4·5-s − 16·7-s + 2·11-s + 28·13-s − 24·15-s − 66·17-s + 66·19-s + 96·21-s − 176·23-s − 184·25-s + 50·27-s − 87·29-s + 190·31-s − 12·33-s − 64·35-s + 442·37-s − 168·39-s + 1.16e3·41-s − 30·43-s + 738·47-s + 39·49-s + 396·51-s + 312·53-s + 8·55-s − 396·57-s − 44·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.357·5-s − 0.863·7-s + 0.0548·11-s + 0.597·13-s − 0.413·15-s − 0.941·17-s + 0.796·19-s + 0.997·21-s − 1.59·23-s − 1.47·25-s + 0.356·27-s − 0.557·29-s + 1.10·31-s − 0.0633·33-s − 0.309·35-s + 1.96·37-s − 0.689·39-s + 4.42·41-s − 0.106·43-s + 2.29·47-s + 0.113·49-s + 1.08·51-s + 0.808·53-s + 0.0196·55-s − 0.920·57-s − 0.0970·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99897344 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99897344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.263151831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263151831\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 p T + 4 p^{2} T^{2} + 166 T^{3} + 4 p^{5} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 8 p^{2} T^{2} - 614 T^{3} + 8 p^{5} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 16 T + 31 p T^{2} - 2480 T^{3} + 31 p^{4} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 3692 T^{2} - 5218 T^{3} + 3692 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 28 T + 2592 T^{2} + 14486 T^{3} + 2592 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 66 T + 14555 T^{2} + 646820 T^{3} + 14555 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 66 T + 10961 T^{2} - 511756 T^{3} + 10961 p^{3} T^{4} - 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 176 T + 44693 T^{2} + 4389472 T^{3} + 44693 p^{3} T^{4} + 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 190 T + 34940 T^{2} - 1815354 T^{3} + 34940 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 442 T + 159351 T^{2} - 34453924 T^{3} + 159351 p^{3} T^{4} - 442 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 1162 T + 645399 T^{2} - 213908268 T^{3} + 645399 p^{3} T^{4} - 1162 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 30 T + 157244 T^{2} + 12805582 T^{3} + 157244 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 738 T + 470756 T^{2} - 163890206 T^{3} + 470756 p^{3} T^{4} - 738 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 312 T + 349160 T^{2} - 97820330 T^{3} + 349160 p^{3} T^{4} - 312 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 44 T + 592857 T^{2} + 18923016 T^{3} + 592857 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 54 T + 121287 T^{2} + 62852868 T^{3} + 121287 p^{3} T^{4} - 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 116 T + 720193 T^{2} - 88800824 T^{3} + 720193 p^{3} T^{4} - 116 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1200 T + 1511489 T^{2} - 903051776 T^{3} + 1511489 p^{3} T^{4} - 1200 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1118 T + 1314795 T^{2} + 850221500 T^{3} + 1314795 p^{3} T^{4} + 1118 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2262 T + 36404 p T^{2} - 2430107338 T^{3} + 36404 p^{4} T^{4} - 2262 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1804 T + 1839409 T^{2} - 1410713352 T^{3} + 1839409 p^{3} T^{4} - 1804 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1578 T + 1639271 T^{2} - 1200535660 T^{3} + 1639271 p^{3} T^{4} - 1578 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1450 T + 1528271 T^{2} - 826334604 T^{3} + 1528271 p^{3} T^{4} - 1450 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.430615798346768330370731753258, −9.177606371934210431711176816796, −9.159465615546279363932806669794, −8.605518140922602634274396435734, −7.943535420446054647030105970367, −7.88084703595962375522281441677, −7.54037264640912237608753512488, −7.49694816305070879236507261126, −6.58516092697133747649306096488, −6.44952664712749111951687501172, −6.22332947313364364843690485431, −5.90842592169912868629374147920, −5.83998349493891296861018088557, −5.35310759435832215862349282416, −5.01252867466951301714540637941, −4.44932930965377188542457333597, −4.04766362039367189696184509248, −3.82247667414419786027880566385, −3.58985184387095464660757613159, −2.51894970812335085969247349872, −2.44077180323286521361124900261, −2.29636980223166907431067628060, −1.13264370554835181153374287161, −0.854832125783655252599219029392, −0.32050615675496514524749856223,
0.32050615675496514524749856223, 0.854832125783655252599219029392, 1.13264370554835181153374287161, 2.29636980223166907431067628060, 2.44077180323286521361124900261, 2.51894970812335085969247349872, 3.58985184387095464660757613159, 3.82247667414419786027880566385, 4.04766362039367189696184509248, 4.44932930965377188542457333597, 5.01252867466951301714540637941, 5.35310759435832215862349282416, 5.83998349493891296861018088557, 5.90842592169912868629374147920, 6.22332947313364364843690485431, 6.44952664712749111951687501172, 6.58516092697133747649306096488, 7.49694816305070879236507261126, 7.54037264640912237608753512488, 7.88084703595962375522281441677, 7.943535420446054647030105970367, 8.605518140922602634274396435734, 9.159465615546279363932806669794, 9.177606371934210431711176816796, 9.430615798346768330370731753258
