| L(s) = 1 | − 2-s + 2.90·3-s + 4-s − 0.347·5-s − 2.90·6-s + 1.00·7-s − 8-s + 5.45·9-s + 0.347·10-s − 5.46·11-s + 2.90·12-s − 6.36·13-s − 1.00·14-s − 1.00·15-s + 16-s − 0.522·17-s − 5.45·18-s − 5.00·19-s − 0.347·20-s + 2.93·21-s + 5.46·22-s − 1.44·23-s − 2.90·24-s − 4.87·25-s + 6.36·26-s + 7.13·27-s + 1.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.67·3-s + 0.5·4-s − 0.155·5-s − 1.18·6-s + 0.381·7-s − 0.353·8-s + 1.81·9-s + 0.109·10-s − 1.64·11-s + 0.839·12-s − 1.76·13-s − 0.269·14-s − 0.260·15-s + 0.250·16-s − 0.126·17-s − 1.28·18-s − 1.14·19-s − 0.0776·20-s + 0.640·21-s + 1.16·22-s − 0.300·23-s − 0.593·24-s − 0.975·25-s + 1.24·26-s + 1.37·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 0.347T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 6.36T + 13T^{2} \) |
| 17 | \( 1 + 0.522T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 6.27T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.0922T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 1.16T + 67T^{2} \) |
| 71 | \( 1 - 0.312T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 0.929T + 97T^{2} \) |
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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
−8.294579680381213552931477563482, −7.83837786699209178051750571103, −7.49391617757707404504396088256, −6.48340259958816754356318395004, −5.14266026048574443367628395823, −4.41110535372813039332571116477, −3.22244879152391783463080653993, −2.41916117983783196815537368722, −1.97989428417860552753909605969, 0,
1.97989428417860552753909605969, 2.41916117983783196815537368722, 3.22244879152391783463080653993, 4.41110535372813039332571116477, 5.14266026048574443367628395823, 6.48340259958816754356318395004, 7.49391617757707404504396088256, 7.83837786699209178051750571103, 8.294579680381213552931477563482
