L(s) = 1 | − 7·2-s − 69·4-s − 250·5-s + 1.30e3·7-s + 413·8-s + 1.75e3·10-s − 3.44e3·11-s − 8.98e3·13-s − 9.12e3·14-s − 4.86e3·16-s + 5.49e3·17-s − 4.95e4·19-s + 1.72e4·20-s + 2.41e4·22-s − 9.18e4·23-s + 4.68e4·25-s + 6.29e4·26-s − 8.99e4·28-s − 1.81e5·29-s + 3.04e5·31-s + 1.61e5·32-s − 3.84e4·34-s − 3.26e5·35-s − 5.02e5·37-s + 3.47e5·38-s − 1.03e5·40-s − 6.31e5·41-s + ⋯ |
L(s) = 1 | − 0.618·2-s − 0.539·4-s − 0.894·5-s + 1.43·7-s + 0.285·8-s + 0.553·10-s − 0.781·11-s − 1.13·13-s − 0.889·14-s − 0.296·16-s + 0.271·17-s − 1.65·19-s + 0.482·20-s + 0.483·22-s − 1.57·23-s + 3/5·25-s + 0.702·26-s − 0.774·28-s − 1.38·29-s + 1.83·31-s + 0.872·32-s − 0.167·34-s − 1.28·35-s − 1.63·37-s + 1.02·38-s − 0.255·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + 7 T + 59 p T^{2} + 7 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1304 T + 1601006 T^{2} - 1304 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3448 T + 9598294 T^{2} + 3448 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8988 T + 43676926 T^{2} + 8988 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5492 T + 533727862 T^{2} - 5492 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 49584 T + 1767259558 T^{2} + 49584 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 91848 T + 4843394254 T^{2} + 91848 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6268 p T + 32439203278 T^{2} + 6268 p^{8} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 502316 T + 221684315886 T^{2} + 502316 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 631172 T + 420346017142 T^{2} + 631172 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 353640 T + 567251429590 T^{2} - 353640 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 467480 T + 1062629128990 T^{2} - 467480 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 568052 T + 2403786403294 T^{2} - 568052 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 287224 T + 1627391637238 T^{2} + 287224 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5073832 T + 16021265240102 T^{2} + 5073832 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3748816 T + 20824804809646 T^{2} - 3748816 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1477212 T - 3158190986 T^{2} + 1477212 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4627720 T + 42789559383518 T^{2} + 4627720 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6072936 T + 62224060120582 T^{2} - 6072936 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16516356 T + 156597055746838 T^{2} + 16516356 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2723428 T + 135845063471622 T^{2} - 2723428 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−13.84188907858769980231030012621, −13.81849805991726470654828166429, −12.64154293168540013826529166892, −12.22829697892357241591768583173, −11.64710674077183346086845416618, −11.03745075735779380247780328528, −10.27130159313228053604390998013, −9.939056423156428691375938383141, −8.734492645305948054844981074791, −8.554147740585570774472803405260, −7.81092460717480919964266114578, −7.47085272984639799998415172378, −6.34083011669719464819863907502, −5.20345335328960321441633260041, −4.61207387956342506577495486947, −4.00798575400841684145448791088, −2.57140154513953339983378026514, −1.62901188374473814017497986832, 0, 0,
1.62901188374473814017497986832, 2.57140154513953339983378026514, 4.00798575400841684145448791088, 4.61207387956342506577495486947, 5.20345335328960321441633260041, 6.34083011669719464819863907502, 7.47085272984639799998415172378, 7.81092460717480919964266114578, 8.554147740585570774472803405260, 8.734492645305948054844981074791, 9.939056423156428691375938383141, 10.27130159313228053604390998013, 11.03745075735779380247780328528, 11.64710674077183346086845416618, 12.22829697892357241591768583173, 12.64154293168540013826529166892, 13.81849805991726470654828166429, 13.84188907858769980231030012621