| L(s) = 1 | − 4.01·2-s + 3·3-s + 8.14·4-s − 12.0·6-s + 1.75·7-s − 0.565·8-s + 9·9-s − 23.3·11-s + 24.4·12-s − 53.2·13-s − 7.06·14-s − 62.8·16-s − 20.8·17-s − 36.1·18-s + 18.3·19-s + 5.27·21-s + 93.8·22-s + 43.0·23-s − 1.69·24-s + 214.·26-s + 27·27-s + 14.3·28-s + 133.·29-s + 242.·31-s + 257.·32-s − 70.1·33-s + 83.9·34-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.01·4-s − 0.820·6-s + 0.0949·7-s − 0.0249·8-s + 0.333·9-s − 0.640·11-s + 0.587·12-s − 1.13·13-s − 0.134·14-s − 0.982·16-s − 0.298·17-s − 0.473·18-s + 0.221·19-s + 0.0548·21-s + 0.909·22-s + 0.390·23-s − 0.0144·24-s + 1.61·26-s + 0.192·27-s + 0.0966·28-s + 0.854·29-s + 1.40·31-s + 1.41·32-s − 0.369·33-s + 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.01T + 8T^{2} \) |
| 7 | \( 1 - 1.75T + 343T^{2} \) |
| 11 | \( 1 + 23.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 43.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 93.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 28.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 451.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 861.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 621.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 687.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 560.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 941.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−8.299610137938379766091592248045, −8.103352826718793160629229313001, −7.16275819960942072123694490003, −6.56570575846625853088376861931, −5.11991258855494968772251916250, −4.42382540569786453077595268295, −2.95289678907784219660141971165, −2.25030664919541813849766625440, −1.10933655632223883562419887699, 0,
1.10933655632223883562419887699, 2.25030664919541813849766625440, 2.95289678907784219660141971165, 4.42382540569786453077595268295, 5.11991258855494968772251916250, 6.56570575846625853088376861931, 7.16275819960942072123694490003, 8.103352826718793160629229313001, 8.299610137938379766091592248045
