| L(s) = 1 | − 0.414·3-s − 5-s − 2·7-s − 2.82·9-s + 2.41·11-s + 1.82·13-s + 0.414·15-s + 2.82·17-s + 7.65·19-s + 0.828·21-s − 1.17·23-s − 4·25-s + 2.41·27-s − 29-s − 2.41·31-s − 0.999·33-s + 2·35-s − 5.65·37-s − 0.757·39-s − 4.82·41-s − 3.24·43-s + 2.82·45-s − 4.41·47-s − 3·49-s − 1.17·51-s − 11.4·53-s − 2.41·55-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 0.447·5-s − 0.755·7-s − 0.942·9-s + 0.727·11-s + 0.507·13-s + 0.106·15-s + 0.685·17-s + 1.75·19-s + 0.180·21-s − 0.244·23-s − 0.800·25-s + 0.464·27-s − 0.185·29-s − 0.433·31-s − 0.174·33-s + 0.338·35-s − 0.929·37-s − 0.121·39-s − 0.754·41-s − 0.494·43-s + 0.421·45-s − 0.643·47-s − 0.428·49-s − 0.164·51-s − 1.57·53-s − 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−8.932344049893036929792596251861, −8.022746840040168230920581407672, −7.31213436258543207603008721930, −6.32728128117044894161727779866, −5.75653570125182153641663014066, −4.82512044295232354882120518762, −3.45575232560401891598105567576, −3.24456624141102292888259492833, −1.49334815356553638844091775082, 0,
1.49334815356553638844091775082, 3.24456624141102292888259492833, 3.45575232560401891598105567576, 4.82512044295232354882120518762, 5.75653570125182153641663014066, 6.32728128117044894161727779866, 7.31213436258543207603008721930, 8.022746840040168230920581407672, 8.932344049893036929792596251861
