| L(s) = 1 | + 2.11·3-s + 3.93·5-s + 1.48·9-s + 4.13·11-s + 1.64·13-s + 8.33·15-s − 5.20·17-s − 6.58·19-s + 7.17·23-s + 10.4·25-s − 3.20·27-s − 9.38·29-s + 6.46·31-s + 8.74·33-s + 11.3·37-s + 3.48·39-s − 41-s − 0.742·43-s + 5.84·45-s + 0.264·47-s − 11.0·51-s − 0.838·53-s + 16.2·55-s − 13.9·57-s − 5.28·59-s + 2.56·61-s + 6.47·65-s + ⋯ |
| L(s) = 1 | + 1.22·3-s + 1.76·5-s + 0.494·9-s + 1.24·11-s + 0.456·13-s + 2.15·15-s − 1.26·17-s − 1.51·19-s + 1.49·23-s + 2.09·25-s − 0.617·27-s − 1.74·29-s + 1.16·31-s + 1.52·33-s + 1.86·37-s + 0.557·39-s − 0.156·41-s − 0.113·43-s + 0.871·45-s + 0.0385·47-s − 1.54·51-s − 0.115·53-s + 2.19·55-s − 1.84·57-s − 0.688·59-s + 0.328·61-s + 0.802·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.077318157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.077318157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.11T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 4.13T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 43 | \( 1 + 0.742T + 43T^{2} \) |
| 47 | \( 1 - 0.264T + 47T^{2} \) |
| 53 | \( 1 + 0.838T + 53T^{2} \) |
| 59 | \( 1 + 5.28T + 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 - 6.32T + 97T^{2} \) |
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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
−8.042018354085624567429060177014, −6.98309536301419965322422219825, −6.39892348905351694088828120413, −6.02413713691096983464293899580, −4.97056262997434465601680717065, −4.19215669731338100283166330954, −3.39121586197161735480121582333, −2.36383734985147367418783203087, −2.10937192480196111993075967523, −1.11903954385246266853464908222,
1.11903954385246266853464908222, 2.10937192480196111993075967523, 2.36383734985147367418783203087, 3.39121586197161735480121582333, 4.19215669731338100283166330954, 4.97056262997434465601680717065, 6.02413713691096983464293899580, 6.39892348905351694088828120413, 6.98309536301419965322422219825, 8.042018354085624567429060177014
