| L(s) = 1 | + 6·3-s + 6·5-s + 16·7-s + 27·9-s + 22·11-s − 22·13-s + 36·15-s − 22·17-s + 62·19-s + 96·21-s − 210·23-s − 86·25-s + 108·27-s + 214·29-s − 328·31-s + 132·33-s + 96·35-s + 200·37-s − 132·39-s + 114·41-s + 606·43-s + 162·45-s − 382·47-s + 54·49-s − 132·51-s + 2·53-s + 132·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.536·5-s + 0.863·7-s + 9-s + 0.603·11-s − 0.469·13-s + 0.619·15-s − 0.313·17-s + 0.748·19-s + 0.997·21-s − 1.90·23-s − 0.687·25-s + 0.769·27-s + 1.37·29-s − 1.90·31-s + 0.696·33-s + 0.463·35-s + 0.888·37-s − 0.541·39-s + 0.434·41-s + 2.14·43-s + 0.536·45-s − 1.18·47-s + 0.157·49-s − 0.362·51-s + 0.00518·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.531386871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.531386871\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22 T + 4378 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 22 T + 8714 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 62 T + 13446 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 210 T + 31934 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 214 T + 53514 T^{2} - 214 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 328 T + 84286 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 200 T + 110758 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 114 T + 25874 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 606 T + 211230 T^{2} - 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 382 T + 183710 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 182538 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1084 T + 703974 T^{2} - 1084 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 354 T + 70866 T^{2} - 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 456 T + 644742 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 34 T + 616238 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 112 T + 109870 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 820 T + 1099378 T^{2} - 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1012 T + 478422 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1036 T + 1658534 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1308 T + 2217990 T^{2} - 1308 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.805197579019903298052845273337, −8.691390330666619776988353971864, −8.004973091705357169018674800717, −7.956135638436507299218624398190, −7.37179475048624066497862817860, −7.30803783341439089386970522770, −6.53447733597239662645567881580, −6.29361699191913687996421467771, −5.60885241545680481729825867610, −5.55892316868867750826187293387, −4.70475854169002389723593222297, −4.52014874033602689998216066198, −3.98115512874609663521150954765, −3.60545760814343358851103562546, −3.14442866706928325847317147571, −2.34892108121895641620031511207, −2.09131936746138517131515805908, −1.87932642002882625364757632663, −1.03293805362742614459922413392, −0.57748589151001023883533360312,
0.57748589151001023883533360312, 1.03293805362742614459922413392, 1.87932642002882625364757632663, 2.09131936746138517131515805908, 2.34892108121895641620031511207, 3.14442866706928325847317147571, 3.60545760814343358851103562546, 3.98115512874609663521150954765, 4.52014874033602689998216066198, 4.70475854169002389723593222297, 5.55892316868867750826187293387, 5.60885241545680481729825867610, 6.29361699191913687996421467771, 6.53447733597239662645567881580, 7.30803783341439089386970522770, 7.37179475048624066497862817860, 7.956135638436507299218624398190, 8.004973091705357169018674800717, 8.691390330666619776988353971864, 8.805197579019903298052845273337
