| L(s) = 1 | − 468·3-s − 5.62e4·5-s + 1.59e6·9-s + 6.39e6·11-s + 3.03e7·13-s + 2.63e7·15-s − 4.31e7·17-s + 3.65e8·19-s + 5.72e7·23-s + 1.22e9·25-s − 2.13e9·27-s − 9.28e7·29-s + 5.68e9·31-s − 2.99e9·33-s + 1.88e9·37-s − 1.42e10·39-s − 1.46e10·41-s − 5.37e10·43-s − 8.96e10·45-s − 1.01e11·47-s + 2.01e10·51-s − 2.78e11·53-s − 3.59e11·55-s − 1.70e11·57-s − 5.95e10·59-s + 2.74e10·61-s − 1.70e12·65-s + ⋯ |
| L(s) = 1 | − 0.370·3-s − 1.60·5-s + 9-s + 1.08·11-s + 1.74·13-s + 0.596·15-s − 0.433·17-s + 1.78·19-s + 0.0806·23-s + 25-s − 1.06·27-s − 0.0289·29-s + 1.14·31-s − 0.403·33-s + 0.120·37-s − 0.647·39-s − 0.482·41-s − 1.29·43-s − 1.60·45-s − 1.37·47-s + 0.160·51-s − 1.72·53-s − 1.75·55-s − 0.659·57-s − 0.183·59-s + 0.0683·61-s − 2.81·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.976972070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976972070\) |
| \(L(\frac{15}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 52 p^{2} T - 16979 p^{4} T^{2} + 52 p^{15} T^{3} + p^{26} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 56214 T + 1939310671 T^{2} + 56214 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 581580 p T + 52923625789 p^{2} T^{2} - 581580 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 15199742 T + p^{13} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 43114194 T - 8045744308636301 T^{2} + 43114194 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 365115484 T + 91256333194297197 T^{2} - 365115484 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 57226824 T - 500761452551340407 T^{2} - 57226824 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 46418994 T + p^{13} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5682185824 T + 7869689441021516385 T^{2} - 5682185824 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1887185098 T - \)\(24\!\cdots\!93\)\( T^{2} - 1887185098 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7336802934 T + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 26886674980 T + p^{13} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 101839834224 T + \)\(49\!\cdots\!49\)\( T^{2} + 101839834224 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 278731884294 T + \)\(51\!\cdots\!63\)\( T^{2} + 278731884294 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 59573945772 T - \)\(10\!\cdots\!95\)\( T^{2} + 59573945772 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 27484470418 T - \)\(16\!\cdots\!57\)\( T^{2} - 27484470418 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 784410054932 T + \)\(67\!\cdots\!37\)\( T^{2} + 784410054932 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360365227992 T + p^{13} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1592635413718 T + \)\(86\!\cdots\!91\)\( T^{2} - 1592635413718 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 23161184752 T - \)\(46\!\cdots\!35\)\( T^{2} - 23161184752 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2050158110436 T + p^{13} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3485391237126 T - \)\(98\!\cdots\!93\)\( T^{2} - 3485391237126 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6706667416802 T + p^{13} T^{2} )^{2} \) |
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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
−10.39175975959297282209815035651, −10.05772979942860355645338792731, −9.260078736914698214567355507465, −9.100742968732378391027613994037, −8.321611803711677235152967810275, −7.77922641591313266127960166255, −7.72368874479589343267241630478, −6.81847520713792590062817560035, −6.53163629025362579850982607400, −6.13415146708623202929821027791, −5.21417823022646221929437995595, −4.77416998538238980650320505853, −4.19838810776529131789834813735, −3.65620849639025240218292073302, −3.52214098308492647886991825373, −2.87717758063919987927101152932, −1.60527647060925818743311800377, −1.47284035229354196644291753345, −0.887892527940874551558263052907, −0.32237145047148958660329505608,
0.32237145047148958660329505608, 0.887892527940874551558263052907, 1.47284035229354196644291753345, 1.60527647060925818743311800377, 2.87717758063919987927101152932, 3.52214098308492647886991825373, 3.65620849639025240218292073302, 4.19838810776529131789834813735, 4.77416998538238980650320505853, 5.21417823022646221929437995595, 6.13415146708623202929821027791, 6.53163629025362579850982607400, 6.81847520713792590062817560035, 7.72368874479589343267241630478, 7.77922641591313266127960166255, 8.321611803711677235152967810275, 9.100742968732378391027613994037, 9.260078736914698214567355507465, 10.05772979942860355645338792731, 10.39175975959297282209815035651
