| L(s) = 1 | + 3.67·3-s + 7.12·5-s − 0.511·7-s − 13.4·9-s − 17.5·11-s − 52.1·13-s + 26.1·15-s + 68.1·17-s + 19·19-s − 1.87·21-s + 48.8·23-s − 74.1·25-s − 148.·27-s + 87.1·29-s + 53.2·31-s − 64.5·33-s − 3.64·35-s + 199.·37-s − 191.·39-s − 372.·41-s − 416.·43-s − 96.2·45-s − 221.·47-s − 342.·49-s + 250.·51-s − 107.·53-s − 125.·55-s + ⋯ |
| L(s) = 1 | + 0.707·3-s + 0.637·5-s − 0.0276·7-s − 0.499·9-s − 0.481·11-s − 1.11·13-s + 0.450·15-s + 0.972·17-s + 0.229·19-s − 0.0195·21-s + 0.442·23-s − 0.593·25-s − 1.06·27-s + 0.558·29-s + 0.308·31-s − 0.340·33-s − 0.0176·35-s + 0.887·37-s − 0.786·39-s − 1.41·41-s − 1.47·43-s − 0.318·45-s − 0.687·47-s − 0.999·49-s + 0.687·51-s − 0.279·53-s − 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 3.67T + 27T^{2} \) |
| 5 | \( 1 - 7.12T + 125T^{2} \) |
| 7 | \( 1 + 0.511T + 343T^{2} \) |
| 11 | \( 1 + 17.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 48.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 416.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 155.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 229.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 38.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 803.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 19.9T + 9.12e5T^{2} \) |
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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
−8.957550149921907284254464152974, −8.123157683996382924971588391338, −7.51185975377632647654411946540, −6.43032272664397148955301807024, −5.49487737567979819295107750206, −4.77214464803016396013513746306, −3.32781679575353325452568889448, −2.68497031115825426144316053675, −1.62703147081652416720007168400, 0,
1.62703147081652416720007168400, 2.68497031115825426144316053675, 3.32781679575353325452568889448, 4.77214464803016396013513746306, 5.49487737567979819295107750206, 6.43032272664397148955301807024, 7.51185975377632647654411946540, 8.123157683996382924971588391338, 8.957550149921907284254464152974
