| L(s) = 1 | + 2-s + 1.62·3-s + 4-s − 1.52·5-s + 1.62·6-s − 1.42·7-s + 8-s − 0.344·9-s − 1.52·10-s − 3.50·11-s + 1.62·12-s − 0.327·13-s − 1.42·14-s − 2.47·15-s + 16-s + 7.06·17-s − 0.344·18-s + 19-s − 1.52·20-s − 2.32·21-s − 3.50·22-s + 3.94·23-s + 1.62·24-s − 2.68·25-s − 0.327·26-s − 5.45·27-s − 1.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.940·3-s + 0.5·4-s − 0.679·5-s + 0.665·6-s − 0.538·7-s + 0.353·8-s − 0.114·9-s − 0.480·10-s − 1.05·11-s + 0.470·12-s − 0.0907·13-s − 0.381·14-s − 0.639·15-s + 0.250·16-s + 1.71·17-s − 0.0812·18-s + 0.229·19-s − 0.339·20-s − 0.506·21-s − 0.746·22-s + 0.822·23-s + 0.332·24-s − 0.537·25-s − 0.0641·26-s − 1.04·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 + 0.327T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 4.95T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 7.45T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 6.38T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.59112457087076986855210917251, −6.97913436680789017311342874912, −5.85289358358215912328450066458, −5.47203934280228171147505011953, −4.54726662622284380581119269131, −3.68170754600413725129144181901, −3.09480827605337953034909348922, −2.72571419847979009497537563750, −1.50487334306441104995165970202, 0,
1.50487334306441104995165970202, 2.72571419847979009497537563750, 3.09480827605337953034909348922, 3.68170754600413725129144181901, 4.54726662622284380581119269131, 5.47203934280228171147505011953, 5.85289358358215912328450066458, 6.97913436680789017311342874912, 7.59112457087076986855210917251
