| L(s) = 1 | + 3.85·3-s − 5·5-s + 23.5·7-s − 12.1·9-s + 58.2·11-s + 68.5·13-s − 19.2·15-s + 101.·17-s + 7.02·19-s + 91.0·21-s + 23·23-s + 25·25-s − 150.·27-s + 206.·29-s − 54.8·31-s + 224.·33-s − 117.·35-s − 241.·37-s + 264.·39-s − 122.·41-s + 320.·43-s + 60.5·45-s − 107.·47-s + 213.·49-s + 391.·51-s + 127.·53-s − 291.·55-s + ⋯ |
| L(s) = 1 | + 0.742·3-s − 0.447·5-s + 1.27·7-s − 0.448·9-s + 1.59·11-s + 1.46·13-s − 0.332·15-s + 1.44·17-s + 0.0848·19-s + 0.945·21-s + 0.208·23-s + 0.200·25-s − 1.07·27-s + 1.31·29-s − 0.317·31-s + 1.18·33-s − 0.569·35-s − 1.07·37-s + 1.08·39-s − 0.466·41-s + 1.13·43-s + 0.200·45-s − 0.332·47-s + 0.622·49-s + 1.07·51-s + 0.329·53-s − 0.713·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.064080566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.064080566\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.85T + 27T^{2} \) |
| 7 | \( 1 - 23.5T + 343T^{2} \) |
| 11 | \( 1 - 58.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.02T + 6.85e3T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 899.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 746.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 903.T + 9.12e5T^{2} \) |
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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
−8.797058606745831140088392132036, −8.174289909244408407027639495521, −7.61916716859479658206386119931, −6.51893529365060437379220459855, −5.70270734101513843354706405195, −4.65310388882829909124670418376, −3.73463372178296237812148345521, −3.15907281046884828556922802084, −1.68267850902723639435096065997, −1.03126694237859618971839368835,
1.03126694237859618971839368835, 1.68267850902723639435096065997, 3.15907281046884828556922802084, 3.73463372178296237812148345521, 4.65310388882829909124670418376, 5.70270734101513843354706405195, 6.51893529365060437379220459855, 7.61916716859479658206386119931, 8.174289909244408407027639495521, 8.797058606745831140088392132036
