| L(s) = 1 | + 2.48·2-s − 3-s + 4.17·4-s − 1.76·5-s − 2.48·6-s + 5.41·8-s + 9-s − 4.39·10-s + 0.293·11-s − 4.17·12-s − 6.33·13-s + 1.76·15-s + 5.11·16-s − 1.00·17-s + 2.48·18-s + 5.87·19-s − 7.38·20-s + 0.729·22-s + 0.838·23-s − 5.41·24-s − 1.87·25-s − 15.7·26-s − 27-s + 2.34·29-s + 4.39·30-s + 1.39·31-s + 1.86·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.08·4-s − 0.790·5-s − 1.01·6-s + 1.91·8-s + 0.333·9-s − 1.38·10-s + 0.0885·11-s − 1.20·12-s − 1.75·13-s + 0.456·15-s + 1.27·16-s − 0.242·17-s + 0.585·18-s + 1.34·19-s − 1.65·20-s + 0.155·22-s + 0.174·23-s − 1.10·24-s − 0.375·25-s − 3.08·26-s − 0.192·27-s + 0.435·29-s + 0.802·30-s + 0.250·31-s + 0.329·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 0.293T + 11T^{2} \) |
| 13 | \( 1 + 6.33T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 0.838T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 + 0.691T + 37T^{2} \) |
| 43 | \( 1 + 0.408T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 4.74T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−7.44923006307106018980020447279, −6.86360711644784593357908082289, −6.12247765878462433014080906966, −5.34132338222179537824878720534, −4.74826267595476918366389938735, −4.34540797976804307642023995971, −3.32303755000726620874919116873, −2.78407122776307682376932950145, −1.62710635073315649232207411308, 0,
1.62710635073315649232207411308, 2.78407122776307682376932950145, 3.32303755000726620874919116873, 4.34540797976804307642023995971, 4.74826267595476918366389938735, 5.34132338222179537824878720534, 6.12247765878462433014080906966, 6.86360711644784593357908082289, 7.44923006307106018980020447279
