| L(s) = 1 | − 2·2-s − 0.334·3-s + 4·4-s + 0.668·6-s + 14.7·7-s − 8·8-s − 26.8·9-s − 6.74·11-s − 1.33·12-s − 79.2·13-s − 29.4·14-s + 16·16-s + 102.·17-s + 53.7·18-s + 116.·19-s − 4.92·21-s + 13.4·22-s + 23·23-s + 2.67·24-s + 158.·26-s + 18.0·27-s + 58.9·28-s + 146.·29-s − 135.·31-s − 32·32-s + 2.25·33-s − 205.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0643·3-s + 0.5·4-s + 0.0455·6-s + 0.795·7-s − 0.353·8-s − 0.995·9-s − 0.184·11-s − 0.0321·12-s − 1.69·13-s − 0.562·14-s + 0.250·16-s + 1.46·17-s + 0.704·18-s + 1.41·19-s − 0.0512·21-s + 0.130·22-s + 0.208·23-s + 0.0227·24-s + 1.19·26-s + 0.128·27-s + 0.397·28-s + 0.940·29-s − 0.786·31-s − 0.176·32-s + 0.0119·33-s − 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 0.334T + 27T^{2} \) |
| 7 | \( 1 - 14.7T + 343T^{2} \) |
| 11 | \( 1 + 6.74T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 146.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 559.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 316.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 246.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 565.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 757.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 434.T + 9.12e5T^{2} \) |
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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
−9.039816309140249636536210165893, −8.055904108663588579386754410727, −7.64316870939333649462111860979, −6.72821888055865436641721207086, −5.30440766566219328157924357402, −5.16521901485155227729630638672, −3.38694757590403112777514542149, −2.51276742404695894139036088932, −1.26886462376550591101505029746, 0,
1.26886462376550591101505029746, 2.51276742404695894139036088932, 3.38694757590403112777514542149, 5.16521901485155227729630638672, 5.30440766566219328157924357402, 6.72821888055865436641721207086, 7.64316870939333649462111860979, 8.055904108663588579386754410727, 9.039816309140249636536210165893
