| L(s) = 1 | − 2.41·3-s + 5-s + 2.82·9-s + 4.82·11-s − 2.82·13-s − 2.41·15-s + 6.82·17-s + 6·19-s + 7.24·23-s + 25-s + 0.414·27-s + 5·29-s + 1.17·31-s − 11.6·33-s + 5.65·37-s + 6.82·39-s − 9.48·41-s + 11.2·43-s + 2.82·45-s + 2·47-s − 16.4·51-s − 3.17·53-s + 4.82·55-s − 14.4·57-s + 3.65·59-s + 3.48·61-s − 2.82·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s + 0.942·9-s + 1.45·11-s − 0.784·13-s − 0.623·15-s + 1.65·17-s + 1.37·19-s + 1.51·23-s + 0.200·25-s + 0.0797·27-s + 0.928·29-s + 0.210·31-s − 2.02·33-s + 0.929·37-s + 1.09·39-s − 1.48·41-s + 1.71·43-s + 0.421·45-s + 0.291·47-s − 2.30·51-s − 0.435·53-s + 0.651·55-s − 1.91·57-s + 0.476·59-s + 0.446·61-s − 0.350·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.844554906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.844554906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 3.48T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 0.414T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.52901400695093342851409230929, −7.07199409820418747969408309685, −6.34126639527717230569085647031, −5.77452529540710926847562618187, −5.13229900284375976767115077203, −4.62810156910857730097144103300, −3.53702519899008268702308149137, −2.75110530584040763158817859957, −1.26341661790949875086895496174, −0.894704853538472539233420699277,
0.894704853538472539233420699277, 1.26341661790949875086895496174, 2.75110530584040763158817859957, 3.53702519899008268702308149137, 4.62810156910857730097144103300, 5.13229900284375976767115077203, 5.77452529540710926847562618187, 6.34126639527717230569085647031, 7.07199409820418747969408309685, 7.52901400695093342851409230929
