| L(s) = 1 | − 0.189·3-s + 1.51·5-s − 2.96·9-s + 5.99·11-s + 2.57·13-s − 0.285·15-s + 2.45·17-s − 7.57·19-s − 4.57·23-s − 2.71·25-s + 1.12·27-s − 7.11·29-s + 6.03·31-s − 1.13·33-s − 4.63·37-s − 0.486·39-s − 41-s − 12.9·43-s − 4.48·45-s + 8.18·47-s − 0.463·51-s − 2.35·53-s + 9.06·55-s + 1.43·57-s − 6.08·59-s − 9.42·61-s + 3.89·65-s + ⋯ |
| L(s) = 1 | − 0.109·3-s + 0.676·5-s − 0.988·9-s + 1.80·11-s + 0.713·13-s − 0.0738·15-s + 0.594·17-s − 1.73·19-s − 0.954·23-s − 0.542·25-s + 0.217·27-s − 1.32·29-s + 1.08·31-s − 0.197·33-s − 0.762·37-s − 0.0779·39-s − 0.156·41-s − 1.96·43-s − 0.668·45-s + 1.19·47-s − 0.0649·51-s − 0.322·53-s + 1.22·55-s + 0.189·57-s − 0.792·59-s − 1.20·61-s + 0.482·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.189T + 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 - 8.18T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 + 9.42T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 0.0799T + 73T^{2} \) |
| 79 | \( 1 - 7.86T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.47096582126485168093793103229, −6.44691187214540153506960117615, −6.17386390755134398133783688214, −5.68784992506809193336017261548, −4.58557452562690562567536837168, −3.87893233366490782683902218274, −3.19900247903349388000085560851, −2.01973904219962935501905225809, −1.45652610264354766190205313993, 0,
1.45652610264354766190205313993, 2.01973904219962935501905225809, 3.19900247903349388000085560851, 3.87893233366490782683902218274, 4.58557452562690562567536837168, 5.68784992506809193336017261548, 6.17386390755134398133783688214, 6.44691187214540153506960117615, 7.47096582126485168093793103229
