L(s) = 1 | + 31.7·2-s + 268.·3-s + 498.·4-s + 8.53e3·6-s − 637.·7-s − 440.·8-s + 5.24e4·9-s − 4.90e4·11-s + 1.33e5·12-s − 7.27e4·13-s − 2.02e4·14-s − 2.69e5·16-s − 6.73e4·17-s + 1.66e6·18-s + 3.41e5·19-s − 1.71e5·21-s − 1.55e6·22-s + 1.34e5·23-s − 1.18e5·24-s − 2.31e6·26-s + 8.81e6·27-s − 3.17e5·28-s + 4.45e6·29-s + 4.56e5·31-s − 8.32e6·32-s − 1.31e7·33-s − 2.13e6·34-s + ⋯ |
L(s) = 1 | + 1.40·2-s + 1.91·3-s + 0.972·4-s + 2.68·6-s − 0.100·7-s − 0.0380·8-s + 2.66·9-s − 1.00·11-s + 1.86·12-s − 0.706·13-s − 0.140·14-s − 1.02·16-s − 0.195·17-s + 3.74·18-s + 0.600·19-s − 0.192·21-s − 1.41·22-s + 0.100·23-s − 0.0727·24-s − 0.991·26-s + 3.19·27-s − 0.0975·28-s + 1.17·29-s + 0.0887·31-s − 1.40·32-s − 1.93·33-s − 0.274·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(5.961292781\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.961292781\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 31.7T + 512T^{2} \) |
| 3 | \( 1 - 268.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 637.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.73e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.34e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.45e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.56e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.30e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.42e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.46e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.78e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.94e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.82e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.85e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.10439484542464571400700767778, −14.05377358745202535926644625964, −13.37969233562154408952193066941, −12.32705528020000392422126122940, −10.03361091542299569749572357312, −8.575300177523225743878026733772, −7.13789794216192565082366786241, −4.85804658318435035742082982376, −3.39096790567837368061594916585, −2.37442858380724106393917883437,
2.37442858380724106393917883437, 3.39096790567837368061594916585, 4.85804658318435035742082982376, 7.13789794216192565082366786241, 8.575300177523225743878026733772, 10.03361091542299569749572357312, 12.32705528020000392422126122940, 13.37969233562154408952193066941, 14.05377358745202535926644625964, 15.10439484542464571400700767778