| L(s) = 1 | + 16·2-s + 201·3-s + 256·4-s + 2.34e3·5-s + 3.21e3·6-s − 8.80e3·7-s + 4.09e3·8-s + 2.07e4·9-s + 3.75e4·10-s − 1.46e4·11-s + 5.14e4·12-s − 1.31e5·13-s − 1.40e5·14-s + 4.72e5·15-s + 6.55e4·16-s − 5.56e4·17-s + 3.31e5·18-s + 1.04e6·19-s + 6.01e5·20-s − 1.77e6·21-s − 2.34e5·22-s − 6.62e5·23-s + 8.23e5·24-s + 3.56e6·25-s − 2.09e6·26-s + 2.08e5·27-s − 2.25e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.43·3-s + 1/2·4-s + 1.68·5-s + 1.01·6-s − 1.38·7-s + 0.353·8-s + 1.05·9-s + 1.18·10-s − 0.301·11-s + 0.716·12-s − 1.27·13-s − 0.980·14-s + 2.40·15-s + 1/4·16-s − 0.161·17-s + 0.744·18-s + 1.83·19-s + 0.840·20-s − 1.98·21-s − 0.213·22-s − 0.493·23-s + 0.506·24-s + 1.82·25-s − 0.899·26-s + 0.0753·27-s − 0.693·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.417233727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.417233727\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 11 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 67 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 2349 T + p^{9} T^{2} \) |
| 7 | \( 1 + 1258 p T + p^{9} T^{2} \) |
| 13 | \( 1 + 131068 T + p^{9} T^{2} \) |
| 17 | \( 1 + 55698 T + p^{9} T^{2} \) |
| 19 | \( 1 - 1041824 T + p^{9} T^{2} \) |
| 23 | \( 1 + 662139 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4819344 T + p^{9} T^{2} \) |
| 31 | \( 1 + 180115 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7803025 T + p^{9} T^{2} \) |
| 41 | \( 1 + 5927736 T + p^{9} T^{2} \) |
| 43 | \( 1 + 5929162 T + p^{9} T^{2} \) |
| 47 | \( 1 - 61576176 T + p^{9} T^{2} \) |
| 53 | \( 1 - 7349514 T + p^{9} T^{2} \) |
| 59 | \( 1 + 113901909 T + p^{9} T^{2} \) |
| 61 | \( 1 + 13814260 T + p^{9} T^{2} \) |
| 67 | \( 1 - 309980903 T + p^{9} T^{2} \) |
| 71 | \( 1 - 42752631 T + p^{9} T^{2} \) |
| 73 | \( 1 - 142018340 T + p^{9} T^{2} \) |
| 79 | \( 1 + 325376446 T + p^{9} T^{2} \) |
| 83 | \( 1 - 253502934 T + p^{9} T^{2} \) |
| 89 | \( 1 + 994227705 T + p^{9} T^{2} \) |
| 97 | \( 1 + 352091047 T + p^{9} T^{2} \) |
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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
−15.52642979465085388380943605669, −14.16672970629116311917968263389, −13.60251750298607132531970420109, −12.61126586229396871945689987976, −9.942593075679538450925461187836, −9.367579651581530442045481828376, −7.19975863869176895085599213833, −5.57951826617949266363777552469, −3.19416377839999206592406108591, −2.18151290644178589601072889475,
2.18151290644178589601072889475, 3.19416377839999206592406108591, 5.57951826617949266363777552469, 7.19975863869176895085599213833, 9.367579651581530442045481828376, 9.942593075679538450925461187836, 12.61126586229396871945689987976, 13.60251750298607132531970420109, 14.16672970629116311917968263389, 15.52642979465085388380943605669
