| L(s) = 1 | + 2-s + 0.938·3-s + 4-s + 0.929·5-s + 0.938·6-s − 1.00·7-s + 8-s − 2.11·9-s + 0.929·10-s + 1.38·11-s + 0.938·12-s − 4.35·13-s − 1.00·14-s + 0.872·15-s + 16-s + 4.33·17-s − 2.11·18-s − 2.75·19-s + 0.929·20-s − 0.946·21-s + 1.38·22-s − 23-s + 0.938·24-s − 4.13·25-s − 4.35·26-s − 4.80·27-s − 1.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.541·3-s + 0.5·4-s + 0.415·5-s + 0.383·6-s − 0.381·7-s + 0.353·8-s − 0.706·9-s + 0.294·10-s + 0.418·11-s + 0.270·12-s − 1.20·13-s − 0.269·14-s + 0.225·15-s + 0.250·16-s + 1.05·17-s − 0.499·18-s − 0.632·19-s + 0.207·20-s − 0.206·21-s + 0.295·22-s − 0.208·23-s + 0.191·24-s − 0.827·25-s − 0.853·26-s − 0.924·27-s − 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.938T + 3T^{2} \) |
| 5 | \( 1 - 0.929T + 5T^{2} \) |
| 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 0.522T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 6.26T + 61T^{2} \) |
| 67 | \( 1 - 8.59T + 67T^{2} \) |
| 71 | \( 1 - 0.963T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 - 3.39T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 9.99T + 97T^{2} \) |
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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
−7.57711334424323026494483888697, −7.06363617806323301521000196464, −6.11187581941031647879133222137, −5.58254704299893573636132012530, −4.93655099280669479676829845321, −3.78615107720438359964769681993, −3.38131756906019302689966815574, −2.37843435251252791011863016510, −1.78387886712264506850217143022, 0,
1.78387886712264506850217143022, 2.37843435251252791011863016510, 3.38131756906019302689966815574, 3.78615107720438359964769681993, 4.93655099280669479676829845321, 5.58254704299893573636132012530, 6.11187581941031647879133222137, 7.06363617806323301521000196464, 7.57711334424323026494483888697
