L(s) = 1 | + 0.560·3-s + 0.521·5-s − 2.68·9-s − 0.305·11-s − 2.33·13-s + 0.291·15-s + 4.89·17-s − 2.13·19-s − 23-s − 4.72·25-s − 3.18·27-s + 8.37·29-s + 7.01·31-s − 0.171·33-s + 1.83·37-s − 1.31·39-s + 2.78·41-s − 9.61·43-s − 1.40·45-s − 4.01·47-s + 2.74·51-s − 5.03·53-s − 0.159·55-s − 1.19·57-s − 1.05·59-s − 9.07·61-s − 1.21·65-s + ⋯ |
L(s) = 1 | + 0.323·3-s + 0.233·5-s − 0.895·9-s − 0.0922·11-s − 0.648·13-s + 0.0753·15-s + 1.18·17-s − 0.490·19-s − 0.208·23-s − 0.945·25-s − 0.612·27-s + 1.55·29-s + 1.25·31-s − 0.0298·33-s + 0.301·37-s − 0.209·39-s + 0.435·41-s − 1.46·43-s − 0.208·45-s − 0.585·47-s + 0.383·51-s − 0.690·53-s − 0.0215·55-s − 0.158·57-s − 0.137·59-s − 1.16·61-s − 0.151·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 0.560T + 3T^{2} \) |
| 5 | \( 1 - 0.521T + 5T^{2} \) |
| 11 | \( 1 + 0.305T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.13T + 19T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 - 7.01T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 0.0980T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 - 0.450T + 97T^{2} \) |
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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
−7.58687964808506833958087609438, −6.56152599156353568639142528831, −6.11825240466498203049104679078, −5.28775315779544158729488141666, −4.70362237317401341526717585496, −3.73462680350087577418795929882, −2.93475427809460737611249886542, −2.38087967285128838674821727281, −1.28031894152658954532599987682, 0,
1.28031894152658954532599987682, 2.38087967285128838674821727281, 2.93475427809460737611249886542, 3.73462680350087577418795929882, 4.70362237317401341526717585496, 5.28775315779544158729488141666, 6.11825240466498203049104679078, 6.56152599156353568639142528831, 7.58687964808506833958087609438