| L(s) = 1 | − 3·3-s + 20.8·5-s + 9·9-s − 15.1·11-s + 2.16·13-s − 62.5·15-s − 119.·17-s + 33.5·19-s − 0.651·23-s + 309.·25-s − 27·27-s − 163.·29-s + 223.·31-s + 45.4·33-s + 168.·37-s − 6.48·39-s − 323.·41-s − 221.·43-s + 187.·45-s − 508.·47-s + 358.·51-s − 176.·53-s − 315.·55-s − 100.·57-s − 454.·59-s + 38.6·61-s + 45.0·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.86·5-s + 0.333·9-s − 0.415·11-s + 0.0461·13-s − 1.07·15-s − 1.70·17-s + 0.404·19-s − 0.00590·23-s + 2.47·25-s − 0.192·27-s − 1.04·29-s + 1.29·31-s + 0.239·33-s + 0.748·37-s − 0.0266·39-s − 1.23·41-s − 0.785·43-s + 0.621·45-s − 1.57·47-s + 0.983·51-s − 0.457·53-s − 0.774·55-s − 0.233·57-s − 1.00·59-s + 0.0811·61-s + 0.0860·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 20.8T + 125T^{2} \) |
| 11 | \( 1 + 15.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.16T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.651T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 508.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 176.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 38.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 602.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 116.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 568.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 383.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−8.424827353068040964449807627030, −7.24875083185844534905649454190, −6.40154637487857140709595476471, −6.06318922165362113280339716025, −5.10109258685485670438334754747, −4.59619421857768661326195573090, −3.07798302522492776500398272227, −2.13553312412437488821494559131, −1.40626396190798642345558716108, 0,
1.40626396190798642345558716108, 2.13553312412437488821494559131, 3.07798302522492776500398272227, 4.59619421857768661326195573090, 5.10109258685485670438334754747, 6.06318922165362113280339716025, 6.40154637487857140709595476471, 7.24875083185844534905649454190, 8.424827353068040964449807627030
