| L(s) = 1 | + 2-s + 2.15·3-s + 4-s + 0.325·5-s + 2.15·6-s + 0.389·7-s + 8-s + 1.65·9-s + 0.325·10-s + 5.83·11-s + 2.15·12-s + 4.16·13-s + 0.389·14-s + 0.701·15-s + 16-s + 4.77·17-s + 1.65·18-s + 3.78·19-s + 0.325·20-s + 0.841·21-s + 5.83·22-s − 23-s + 2.15·24-s − 4.89·25-s + 4.16·26-s − 2.90·27-s + 0.389·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.24·3-s + 0.5·4-s + 0.145·5-s + 0.880·6-s + 0.147·7-s + 0.353·8-s + 0.550·9-s + 0.102·10-s + 1.75·11-s + 0.622·12-s + 1.15·13-s + 0.104·14-s + 0.181·15-s + 0.250·16-s + 1.15·17-s + 0.389·18-s + 0.868·19-s + 0.0727·20-s + 0.183·21-s + 1.24·22-s − 0.208·23-s + 0.440·24-s − 0.978·25-s + 0.816·26-s − 0.559·27-s + 0.0737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.433814159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.433814159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 - 0.325T + 5T^{2} \) |
| 7 | \( 1 - 0.389T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 5.97T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 + 9.88T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.314T + 71T^{2} \) |
| 73 | \( 1 - 0.0460T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.093239946774087981691837630586, −7.40168204930509116896342655142, −6.63528605901246554704457051559, −5.93052046843747424112224720491, −5.22166570361144819197047240205, −4.07237147596028072328275552093, −3.55409881760856378494664994382, −3.16439575579093087939160978107, −1.85006150321293159702176274445, −1.33922733204956519725505337289,
1.33922733204956519725505337289, 1.85006150321293159702176274445, 3.16439575579093087939160978107, 3.55409881760856378494664994382, 4.07237147596028072328275552093, 5.22166570361144819197047240205, 5.93052046843747424112224720491, 6.63528605901246554704457051559, 7.40168204930509116896342655142, 8.093239946774087981691837630586
