| L(s) = 1 | + 0.535·2-s + 3-s − 1.71·4-s + 0.535·6-s + 3.02·7-s − 1.98·8-s + 9-s + 4.89·11-s − 1.71·12-s − 6.47·13-s + 1.61·14-s + 2.36·16-s + 5.64·17-s + 0.535·18-s + 6.85·19-s + 3.02·21-s + 2.62·22-s + 5.10·23-s − 1.98·24-s − 3.46·26-s + 27-s − 5.17·28-s + 1.07·29-s + 6.24·31-s + 5.24·32-s + 4.89·33-s + 3.02·34-s + ⋯ |
| L(s) = 1 | + 0.378·2-s + 0.577·3-s − 0.856·4-s + 0.218·6-s + 1.14·7-s − 0.702·8-s + 0.333·9-s + 1.47·11-s − 0.494·12-s − 1.79·13-s + 0.432·14-s + 0.590·16-s + 1.37·17-s + 0.126·18-s + 1.57·19-s + 0.659·21-s + 0.558·22-s + 1.06·23-s − 0.405·24-s − 0.679·26-s + 0.192·27-s − 0.978·28-s + 0.198·29-s + 1.12·31-s + 0.926·32-s + 0.852·33-s + 0.518·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.356639407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356639407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 - 0.535T + 2T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 + 0.490T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 - 0.248T + 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 + 4.40T + 97T^{2} \) |
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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
−7.933691009783994179832665438643, −7.26120400686732881314859086778, −6.52780016345721753485029709956, −5.33540993651379198016105606118, −4.99414631479190935015794267661, −4.44136013554414937033629745423, −3.42832489028168820912768875495, −2.99376191888875304715168366199, −1.66439403171706365197912167070, −0.919158196407086784420353477187,
0.919158196407086784420353477187, 1.66439403171706365197912167070, 2.99376191888875304715168366199, 3.42832489028168820912768875495, 4.44136013554414937033629745423, 4.99414631479190935015794267661, 5.33540993651379198016105606118, 6.52780016345721753485029709956, 7.26120400686732881314859086778, 7.933691009783994179832665438643
