| L(s) = 1 | − 2-s − 3.07·3-s + 4-s + 0.557·5-s + 3.07·6-s + 3.52·7-s − 8-s + 6.47·9-s − 0.557·10-s − 0.442·11-s − 3.07·12-s − 2.71·13-s − 3.52·14-s − 1.71·15-s + 16-s + 17-s − 6.47·18-s − 6.31·19-s + 0.557·20-s − 10.8·21-s + 0.442·22-s + 3.08·23-s + 3.07·24-s − 4.68·25-s + 2.71·26-s − 10.6·27-s + 3.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s + 0.249·5-s + 1.25·6-s + 1.33·7-s − 0.353·8-s + 2.15·9-s − 0.176·10-s − 0.133·11-s − 0.888·12-s − 0.753·13-s − 0.940·14-s − 0.443·15-s + 0.250·16-s + 0.242·17-s − 1.52·18-s − 1.44·19-s + 0.124·20-s − 2.36·21-s + 0.0943·22-s + 0.643·23-s + 0.628·24-s − 0.937·25-s + 0.532·26-s − 2.05·27-s + 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.557T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 0.442T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + 6.61T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 - 9.14T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.819781915423211279059497517518, −7.79162824439744256410926421346, −7.25785880271422905298853355695, −6.31981021542493210321973900097, −5.63288203280975822076025679227, −4.91666130345524052310623396020, −4.16621855673730576068218133474, −2.24249258489959007095725414229, −1.31745012264088834612861753725, 0,
1.31745012264088834612861753725, 2.24249258489959007095725414229, 4.16621855673730576068218133474, 4.91666130345524052310623396020, 5.63288203280975822076025679227, 6.31981021542493210321973900097, 7.25785880271422905298853355695, 7.79162824439744256410926421346, 8.819781915423211279059497517518
