| L(s) = 1 | + 1.58·3-s + 3.10·5-s − 2.36·7-s − 0.472·9-s + 0.816·11-s − 13-s + 4.93·15-s − 3.91·17-s − 3.83·19-s − 3.75·21-s + 0.187·23-s + 4.65·25-s − 5.52·27-s − 29-s − 5.86·31-s + 1.29·33-s − 7.33·35-s − 3.70·37-s − 1.58·39-s + 6.03·41-s − 7.79·43-s − 1.46·45-s − 12.8·47-s − 1.42·49-s − 6.22·51-s + 0.553·53-s + 2.53·55-s + ⋯ |
| L(s) = 1 | + 0.917·3-s + 1.38·5-s − 0.892·7-s − 0.157·9-s + 0.246·11-s − 0.277·13-s + 1.27·15-s − 0.949·17-s − 0.878·19-s − 0.818·21-s + 0.0390·23-s + 0.930·25-s − 1.06·27-s − 0.185·29-s − 1.05·31-s + 0.225·33-s − 1.23·35-s − 0.609·37-s − 0.254·39-s + 0.941·41-s − 1.18·43-s − 0.218·45-s − 1.87·47-s − 0.204·49-s − 0.871·51-s + 0.0759·53-s + 0.341·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 - 0.816T + 11T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 0.187T + 23T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 0.553T + 53T^{2} \) |
| 59 | \( 1 + 1.22T + 59T^{2} \) |
| 61 | \( 1 - 0.251T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83248128149098474129078810320, −6.79973779840551898264373804407, −6.42478389970564920453277702959, −5.68504780213637351088460669937, −4.89070103038456453103557025678, −3.82263245503824882371911753584, −3.09991738146704475396624792261, −2.27400866679591937776133653694, −1.76568692246176394110572440888, 0,
1.76568692246176394110572440888, 2.27400866679591937776133653694, 3.09991738146704475396624792261, 3.82263245503824882371911753584, 4.89070103038456453103557025678, 5.68504780213637351088460669937, 6.42478389970564920453277702959, 6.79973779840551898264373804407, 7.83248128149098474129078810320
