| L(s) = 1 | − 1.97·2-s + 1.91·4-s + 2.84·5-s − 3.61·7-s + 0.176·8-s − 5.62·10-s + 2.43·11-s + 2.04·13-s + 7.14·14-s − 4.17·16-s − 3.16·17-s − 19-s + 5.43·20-s − 4.82·22-s − 7.32·23-s + 3.08·25-s − 4.03·26-s − 6.90·28-s + 8.41·29-s − 2.73·31-s + 7.89·32-s + 6.25·34-s − 10.2·35-s + 8.35·37-s + 1.97·38-s + 0.500·40-s + 0.282·41-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.955·4-s + 1.27·5-s − 1.36·7-s + 0.0622·8-s − 1.77·10-s + 0.734·11-s + 0.566·13-s + 1.90·14-s − 1.04·16-s − 0.767·17-s − 0.229·19-s + 1.21·20-s − 1.02·22-s − 1.52·23-s + 0.617·25-s − 0.791·26-s − 1.30·28-s + 1.56·29-s − 0.490·31-s + 1.39·32-s + 1.07·34-s − 1.73·35-s + 1.37·37-s + 0.320·38-s + 0.0791·40-s + 0.0441·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.97T + 2T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 0.282T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 + 0.822T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 + 8.03T + 67T^{2} \) |
| 71 | \( 1 + 2.25T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−7.60846061071367546089901932438, −6.56743896461808436133131664124, −6.45184192243018089188688921917, −5.84139746972435379672754487295, −4.62633542906129928523272233990, −3.80970188534539769765486589424, −2.72769611296228099810914209076, −1.99821113138486588241428081735, −1.14932070684685058650734054900, 0,
1.14932070684685058650734054900, 1.99821113138486588241428081735, 2.72769611296228099810914209076, 3.80970188534539769765486589424, 4.62633542906129928523272233990, 5.84139746972435379672754487295, 6.45184192243018089188688921917, 6.56743896461808436133131664124, 7.60846061071367546089901932438
