| L(s) = 1 | + 7.95·3-s − 5·5-s − 21.6·7-s + 36.3·9-s − 55.3·11-s + 78.8·13-s − 39.7·15-s + 21.6·17-s + 62.8·19-s − 172.·21-s + 23·23-s + 25·25-s + 74.1·27-s + 150.·29-s − 186.·31-s − 440.·33-s + 108.·35-s − 255.·37-s + 627.·39-s − 232.·41-s − 200.·43-s − 181.·45-s + 393.·47-s + 127.·49-s + 172.·51-s − 45.9·53-s + 276.·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.447·5-s − 1.17·7-s + 1.34·9-s − 1.51·11-s + 1.68·13-s − 0.684·15-s + 0.309·17-s + 0.758·19-s − 1.79·21-s + 0.208·23-s + 0.200·25-s + 0.528·27-s + 0.960·29-s − 1.08·31-s − 2.32·33-s + 0.523·35-s − 1.13·37-s + 2.57·39-s − 0.887·41-s − 0.712·43-s − 0.601·45-s + 1.22·47-s + 0.371·49-s + 0.473·51-s − 0.119·53-s + 0.678·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7.95T + 27T^{2} \) |
| 7 | \( 1 + 21.6T + 343T^{2} \) |
| 11 | \( 1 + 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 255.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 200.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 45.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 307.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 538.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 424.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 105.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 941.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 983.T + 9.12e5T^{2} \) |
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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
−8.659299442190487104621756890232, −7.78644641936436409703137080418, −7.22007227967247294492459149587, −6.19399116659544892982830074411, −5.21544993472761822056039446813, −3.91732725532252822368860385275, −3.24384851767838902696436061558, −2.81588136020400253106772714907, −1.46635957609314482884607517528, 0,
1.46635957609314482884607517528, 2.81588136020400253106772714907, 3.24384851767838902696436061558, 3.91732725532252822368860385275, 5.21544993472761822056039446813, 6.19399116659544892982830074411, 7.22007227967247294492459149587, 7.78644641936436409703137080418, 8.659299442190487104621756890232
