| L(s) = 1 | + 1.16·2-s − 1.75·3-s − 0.631·4-s − 2.05·6-s − 5.24·7-s − 3.07·8-s + 0.0891·9-s + 6.37·11-s + 1.11·12-s + 3.65·13-s − 6.13·14-s − 2.33·16-s − 1.90·17-s + 0.104·18-s + 2.01·19-s + 9.21·21-s + 7.45·22-s + 6.73·23-s + 5.41·24-s + 4.27·26-s + 5.11·27-s + 3.31·28-s − 2.61·29-s + 2.74·31-s + 3.42·32-s − 11.2·33-s − 2.22·34-s + ⋯ |
| L(s) = 1 | + 0.827·2-s − 1.01·3-s − 0.315·4-s − 0.839·6-s − 1.98·7-s − 1.08·8-s + 0.0297·9-s + 1.92·11-s + 0.320·12-s + 1.01·13-s − 1.63·14-s − 0.584·16-s − 0.461·17-s + 0.0245·18-s + 0.462·19-s + 2.01·21-s + 1.58·22-s + 1.40·23-s + 1.10·24-s + 0.837·26-s + 0.984·27-s + 0.625·28-s − 0.485·29-s + 0.492·31-s + 0.605·32-s − 1.95·33-s − 0.382·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078073083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078073083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 3.65T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 41 | \( 1 - 0.182T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + 7.98T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 - 2.56T + 73T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 - 9.64T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−10.04084893556417460216399971928, −9.165540063992267194583427035028, −8.851894119497669079663150364882, −6.89503804280205607841421130157, −6.32375403360632977037993562052, −5.95890914530078382431254617589, −4.79498922616113773922161605384, −3.71594131708635787740246745458, −3.17530379923579738147672256614, −0.76523694948527899187525107900,
0.76523694948527899187525107900, 3.17530379923579738147672256614, 3.71594131708635787740246745458, 4.79498922616113773922161605384, 5.95890914530078382431254617589, 6.32375403360632977037993562052, 6.89503804280205607841421130157, 8.851894119497669079663150364882, 9.165540063992267194583427035028, 10.04084893556417460216399971928
