| L(s) = 1 | + 6·3-s − 14·5-s + 27·9-s − 18·11-s − 48·13-s − 84·15-s − 34·17-s − 16·19-s − 110·23-s + 74·25-s + 108·27-s + 212·29-s − 136·31-s − 108·33-s − 24·37-s − 288·39-s − 694·41-s + 584·43-s − 378·45-s − 316·47-s − 204·51-s + 560·53-s + 252·55-s − 96·57-s − 492·59-s + 604·61-s + 672·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.25·5-s + 9-s − 0.493·11-s − 1.02·13-s − 1.44·15-s − 0.485·17-s − 0.193·19-s − 0.997·23-s + 0.591·25-s + 0.769·27-s + 1.35·29-s − 0.787·31-s − 0.569·33-s − 0.106·37-s − 1.18·39-s − 2.64·41-s + 2.07·43-s − 1.25·45-s − 0.980·47-s − 0.560·51-s + 1.45·53-s + 0.617·55-s − 0.223·57-s − 1.08·59-s + 1.26·61-s + 1.28·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.370399596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.370399596\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 18 T + 1150 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 48 T + 4262 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 p T - 4222 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 16 T + 10950 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 110 T + 18686 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 212 T + 57182 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 136 T + 61374 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 694 T + 243914 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 316 T + 231902 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 560 T + 369782 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 492 T + 351622 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 604 T + 406398 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1710 T + 1336222 T^{2} - 1710 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1312 T + 1201998 T^{2} - 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 556 T + 751134 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 264 T + 979750 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 70 T - 186262 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 136 T + 1812270 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.575538226071418621300594096551, −8.306989345462196982494481588068, −8.058125748457096858027425184698, −7.987728412473259123450125143397, −7.20771883589076009047012964608, −7.13301480205609200915360921693, −6.64941730388337690770852370244, −6.36756352232415575144649635388, −5.42609436563531896250073384390, −5.28959162392980138117898220123, −4.68700379603590535584730063871, −4.39299688390228512666611454646, −3.72097414335759003982034198056, −3.71445667165639460204541437344, −3.11133454528342087656805183134, −2.52315834012646762498768932446, −2.15858186618317234552972511798, −1.79623628339182733818883260093, −0.66564183129699699194764421106, −0.49064354949624255040392876559,
0.49064354949624255040392876559, 0.66564183129699699194764421106, 1.79623628339182733818883260093, 2.15858186618317234552972511798, 2.52315834012646762498768932446, 3.11133454528342087656805183134, 3.71445667165639460204541437344, 3.72097414335759003982034198056, 4.39299688390228512666611454646, 4.68700379603590535584730063871, 5.28959162392980138117898220123, 5.42609436563531896250073384390, 6.36756352232415575144649635388, 6.64941730388337690770852370244, 7.13301480205609200915360921693, 7.20771883589076009047012964608, 7.987728412473259123450125143397, 8.058125748457096858027425184698, 8.306989345462196982494481588068, 8.575538226071418621300594096551
