| L(s) = 1 | − 2.30·2-s + 3.30·4-s + 5-s + 7-s − 3.00·8-s − 2.30·10-s − 3.60·13-s − 2.30·14-s + 0.302·16-s − 4·17-s − 3·19-s + 3.30·20-s + 2·23-s − 4·25-s + 8.30·26-s + 3.30·28-s + 5.60·29-s − 2·31-s + 5.30·32-s + 9.21·34-s + 35-s − 8.21·37-s + 6.90·38-s − 3.00·40-s + 7.21·41-s + 5.21·43-s − 4.60·46-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s + 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.728·10-s − 1.00·13-s − 0.615·14-s + 0.0756·16-s − 0.970·17-s − 0.688·19-s + 0.738·20-s + 0.417·23-s − 0.800·25-s + 1.62·26-s + 0.624·28-s + 1.04·29-s − 0.359·31-s + 0.937·32-s + 1.57·34-s + 0.169·35-s − 1.34·37-s + 1.12·38-s − 0.474·40-s + 1.12·41-s + 0.794·43-s − 0.679·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 - 2.39T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 3.21T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−7.68805314585137201209147138996, −6.96348245816583632227834691761, −6.55838305088936846269708326986, −5.55658633624942860197561097025, −4.77539638602473578594741092430, −3.90484552970745247985954111918, −2.40617129225806622154369612368, −2.23846994650785783002371243923, −1.08672684546373616739928384281, 0,
1.08672684546373616739928384281, 2.23846994650785783002371243923, 2.40617129225806622154369612368, 3.90484552970745247985954111918, 4.77539638602473578594741092430, 5.55658633624942860197561097025, 6.55838305088936846269708326986, 6.96348245816583632227834691761, 7.68805314585137201209147138996
