| L(s) = 1 | + 1.19·3-s + 1.56·5-s + 3.07·7-s − 1.56·9-s − 5.47·11-s + 1.87·15-s − 2.12·17-s − 7.34·19-s + 3.68·21-s + 3.59·23-s − 2.56·25-s − 5.47·27-s − 5·29-s + 6.67·31-s − 6.56·33-s + 4.79·35-s − 6.12·37-s + 4.12·41-s + 0.673·43-s − 2.43·45-s − 0.525·47-s + 2.43·49-s − 2.54·51-s + 1.56·53-s − 8.54·55-s − 8.80·57-s − 10.2·59-s + ⋯ |
| L(s) = 1 | + 0.692·3-s + 0.698·5-s + 1.16·7-s − 0.520·9-s − 1.64·11-s + 0.483·15-s − 0.514·17-s − 1.68·19-s + 0.804·21-s + 0.750·23-s − 0.512·25-s − 1.05·27-s − 0.928·29-s + 1.19·31-s − 1.14·33-s + 0.810·35-s − 1.00·37-s + 0.643·41-s + 0.102·43-s − 0.363·45-s − 0.0767·47-s + 0.348·49-s − 0.356·51-s + 0.214·53-s − 1.15·55-s − 1.16·57-s − 1.33·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 0.673T + 43T^{2} \) |
| 47 | \( 1 + 0.525T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.892255953203760998988277445819, −7.37610362345820856618107535584, −6.25570976595875530790603014563, −5.60708556193876693146679913359, −4.91464501094266458922719555723, −4.20876365295922114832546040960, −2.98162221951228845206230838659, −2.33414751346608463906200025035, −1.73490746939009730627409484555, 0,
1.73490746939009730627409484555, 2.33414751346608463906200025035, 2.98162221951228845206230838659, 4.20876365295922114832546040960, 4.91464501094266458922719555723, 5.60708556193876693146679913359, 6.25570976595875530790603014563, 7.37610362345820856618107535584, 7.892255953203760998988277445819
