| L(s) = 1 | + 16·2-s + 21·3-s + 192·4-s − 413·5-s + 336·6-s + 686·7-s + 2.04e3·8-s − 3.03e3·9-s − 6.60e3·10-s + 2.66e3·11-s + 4.03e3·12-s − 5.72e3·13-s + 1.09e4·14-s − 8.67e3·15-s + 2.04e4·16-s − 2.36e3·17-s − 4.85e4·18-s − 1.56e4·19-s − 7.92e4·20-s + 1.44e4·21-s + 4.25e4·22-s − 3.75e4·23-s + 4.30e4·24-s − 1.47e4·25-s − 9.16e4·26-s − 9.07e4·27-s + 1.31e5·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.449·3-s + 3/2·4-s − 1.47·5-s + 0.635·6-s + 0.755·7-s + 1.41·8-s − 1.38·9-s − 2.08·10-s + 0.603·11-s + 0.673·12-s − 0.722·13-s + 1.06·14-s − 0.663·15-s + 5/4·16-s − 0.116·17-s − 1.96·18-s − 0.523·19-s − 2.21·20-s + 0.339·21-s + 0.852·22-s − 0.643·23-s + 0.635·24-s − 0.188·25-s − 1.02·26-s − 0.887·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 7 p T + 386 p^{2} T^{2} - 7 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 413 T + 37062 p T^{2} + 413 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5726 T + 35771842 T^{2} + 5726 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2366 T + 815718546 T^{2} + 2366 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 15652 T + 1564496170 T^{2} + 15652 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 37541 T + 7012238258 T^{2} + 37541 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 164208 T + 29127860870 T^{2} + 164208 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 119273 T + 39876144312 T^{2} + 119273 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 517723 T + 199169242168 T^{2} + 517723 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 443170 T + 428668935762 T^{2} + 443170 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 382680 T + 555899623110 T^{2} + 382680 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 366674 T + 1041762474206 T^{2} + 366674 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 369464 T + 410478710598 T^{2} + 369464 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 775607 T + 3950461055958 T^{2} - 775607 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 309708 T + 3809100555534 T^{2} - 309708 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6769843 T + 23186849804802 T^{2} + 6769843 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4942531 T + 19037878820790 T^{2} + 4942531 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3473134 T + 22770350650882 T^{2} + 3473134 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2066106 T + 39469883881838 T^{2} + 2066106 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4602514 T + 44658086650782 T^{2} + 4602514 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7529515 T + 99213521624108 T^{2} - 7529515 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20463205 T + 250744729268272 T^{2} - 20463205 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.62022467291813457911919702773, −11.32982111601783664047832735639, −10.55965129064916264675834604121, −10.15585120324282200429147802357, −8.935706458729627690517364101235, −8.869262955489605742617674215011, −7.917654615969877722409245907260, −7.81317420081086206373458948891, −7.17346348613128858414279747223, −6.53198945172018454007503989698, −5.67948100909831623287787464062, −5.41232680931941986887181840344, −4.46971797757819344921556305927, −4.20670622655199017144012044932, −3.30798942299883007286007726909, −3.25401980540532905835705264790, −2.05376772191971197117028153972, −1.71412309833694170492545425824, 0, 0,
1.71412309833694170492545425824, 2.05376772191971197117028153972, 3.25401980540532905835705264790, 3.30798942299883007286007726909, 4.20670622655199017144012044932, 4.46971797757819344921556305927, 5.41232680931941986887181840344, 5.67948100909831623287787464062, 6.53198945172018454007503989698, 7.17346348613128858414279747223, 7.81317420081086206373458948891, 7.917654615969877722409245907260, 8.869262955489605742617674215011, 8.935706458729627690517364101235, 10.15585120324282200429147802357, 10.55965129064916264675834604121, 11.32982111601783664047832735639, 11.62022467291813457911919702773
