| L(s) = 1 | − 1.78·2-s + 0.315·3-s + 1.19·4-s − 5-s − 0.563·6-s − 7-s + 1.43·8-s − 2.90·9-s + 1.78·10-s + 0.377·12-s + 0.918·13-s + 1.78·14-s − 0.315·15-s − 4.96·16-s − 2.10·17-s + 5.18·18-s − 1.48·19-s − 1.19·20-s − 0.315·21-s + 7.48·23-s + 0.452·24-s + 25-s − 1.64·26-s − 1.85·27-s − 1.19·28-s − 4.91·29-s + 0.563·30-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.181·3-s + 0.598·4-s − 0.447·5-s − 0.229·6-s − 0.377·7-s + 0.507·8-s − 0.966·9-s + 0.565·10-s + 0.108·12-s + 0.254·13-s + 0.477·14-s − 0.0813·15-s − 1.24·16-s − 0.509·17-s + 1.22·18-s − 0.339·19-s − 0.267·20-s − 0.0687·21-s + 1.56·23-s + 0.0923·24-s + 0.200·25-s − 0.322·26-s − 0.357·27-s − 0.226·28-s − 0.912·29-s + 0.102·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 - 0.315T + 3T^{2} \) |
| 13 | \( 1 - 0.918T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 0.211T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 - 8.80T + 53T^{2} \) |
| 59 | \( 1 - 0.639T + 59T^{2} \) |
| 61 | \( 1 - 3.61T + 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.261372811909940545860533569526, −7.49353886529767873496822392946, −6.84697999616508390634821481475, −6.04933151995271402551599358482, −5.02640317543936541901478380269, −4.18948378775781955689952478298, −3.15965163361589608734106131985, −2.34153001450534797169332430392, −1.05359836245536826353664117478, 0,
1.05359836245536826353664117478, 2.34153001450534797169332430392, 3.15965163361589608734106131985, 4.18948378775781955689952478298, 5.02640317543936541901478380269, 6.04933151995271402551599358482, 6.84697999616508390634821481475, 7.49353886529767873496822392946, 8.261372811909940545860533569526
