| L(s) = 1 | + 16·2-s + 256·4-s + 2.01e3·5-s − 4.56e3·7-s + 4.09e3·8-s + 3.22e4·10-s + 6.98e4·11-s + 4.56e4·13-s − 7.30e4·14-s + 6.55e4·16-s − 1.42e5·17-s − 7.59e5·19-s + 5.15e5·20-s + 1.11e6·22-s + 1.87e6·23-s + 2.10e6·25-s + 7.29e5·26-s − 1.16e6·28-s + 7.97e5·29-s + 8.93e6·31-s + 1.04e6·32-s − 2.28e6·34-s − 9.19e6·35-s + 1.97e7·37-s − 1.21e7·38-s + 8.24e6·40-s − 1.92e7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.718·7-s + 0.353·8-s + 1.01·10-s + 1.43·11-s + 0.442·13-s − 0.508·14-s + 0.250·16-s − 0.414·17-s − 1.33·19-s + 0.720·20-s + 1.01·22-s + 1.39·23-s + 1.07·25-s + 0.313·26-s − 0.359·28-s + 0.209·29-s + 1.73·31-s + 0.176·32-s − 0.293·34-s − 1.03·35-s + 1.73·37-s − 0.945·38-s + 0.509·40-s − 1.06·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.053522354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.053522354\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.01e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.56e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.42e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.59e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.87e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.97e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.93e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.97e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.38e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.81e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.17e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.42e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.08e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.23e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.43e8T + 7.60e17T^{2} \) |
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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
−13.42910313610805054471631145817, −12.61579071822473890653255099794, −11.16331253713897158500470427110, −9.892316094368056645130882014312, −8.875961601017031369802998468095, −6.59869994465647640274398232281, −6.12200008592349972222008400048, −4.42632705215302090642406627922, −2.78647977091706962784863956493, −1.34456264274971010457777539775,
1.34456264274971010457777539775, 2.78647977091706962784863956493, 4.42632705215302090642406627922, 6.12200008592349972222008400048, 6.59869994465647640274398232281, 8.875961601017031369802998468095, 9.892316094368056645130882014312, 11.16331253713897158500470427110, 12.61579071822473890653255099794, 13.42910313610805054471631145817
