| L(s) = 1 | + 0.772·2-s − 3.03·3-s − 1.40·4-s + 5-s − 2.34·6-s + 1.44·7-s − 2.62·8-s + 6.20·9-s + 0.772·10-s + 11-s + 4.25·12-s + 1.66·13-s + 1.11·14-s − 3.03·15-s + 0.775·16-s − 5.51·17-s + 4.79·18-s − 7.79·19-s − 1.40·20-s − 4.39·21-s + 0.772·22-s + 1.35·23-s + 7.97·24-s + 25-s + 1.28·26-s − 9.72·27-s − 2.03·28-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 1.75·3-s − 0.701·4-s + 0.447·5-s − 0.956·6-s + 0.547·7-s − 0.929·8-s + 2.06·9-s + 0.244·10-s + 0.301·11-s + 1.22·12-s + 0.461·13-s + 0.299·14-s − 0.783·15-s + 0.193·16-s − 1.33·17-s + 1.12·18-s − 1.78·19-s − 0.313·20-s − 0.959·21-s + 0.164·22-s + 0.282·23-s + 1.62·24-s + 0.200·25-s + 0.252·26-s − 1.87·27-s − 0.384·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.772T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 0.777T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 79 | \( 1 + 4.18T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 + 2.41T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.216608269122972182083116787763, −6.79860164219741134557031436033, −6.44263096549835991051962650028, −5.79885256731821180352039404214, −5.04723159586216371249564588647, −4.46345864664084886248893836716, −3.97709228015042327904231039893, −2.37216411986671366992442218254, −1.14762255864249337265289334382, 0,
1.14762255864249337265289334382, 2.37216411986671366992442218254, 3.97709228015042327904231039893, 4.46345864664084886248893836716, 5.04723159586216371249564588647, 5.79885256731821180352039404214, 6.44263096549835991051962650028, 6.79860164219741134557031436033, 8.216608269122972182083116787763
