L(s) = 1 | + 0.618·3-s + 4.94·7-s − 2.61·9-s − 1.71·11-s − 6.00·13-s − 6.56·17-s + 4.62·19-s + 3.05·21-s + 8.18·23-s − 3.47·27-s − 5.23·29-s − 4.47·31-s − 1.05·33-s + 3.53·37-s − 3.71·39-s − 3.43·41-s + 10.0·43-s − 5.05·47-s + 17.4·49-s − 4.05·51-s + 8.47·53-s + 2.85·57-s + 2.67·59-s − 4.95·61-s − 12.9·63-s + 1.67·67-s + 5.05·69-s + ⋯ |
L(s) = 1 | + 0.356·3-s + 1.87·7-s − 0.872·9-s − 0.516·11-s − 1.66·13-s − 1.59·17-s + 1.06·19-s + 0.667·21-s + 1.70·23-s − 0.668·27-s − 0.972·29-s − 0.803·31-s − 0.184·33-s + 0.580·37-s − 0.594·39-s − 0.537·41-s + 1.53·43-s − 0.737·47-s + 2.49·49-s − 0.568·51-s + 1.16·53-s + 0.378·57-s + 0.347·59-s − 0.634·61-s − 1.63·63-s + 0.204·67-s + 0.608·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.67T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 1.32T + 73T^{2} \) |
| 79 | \( 1 - 6.00T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−7.47623910592486247494257868690, −6.92871663068856734968628256461, −5.65820857910524980839070467980, −5.14313241598963574828292593702, −4.77099426953307142103750450811, −3.88399987889765925427574582877, −2.61816352023936586116364193759, −2.41840197817906227223670620218, −1.34938684344712176569677399158, 0,
1.34938684344712176569677399158, 2.41840197817906227223670620218, 2.61816352023936586116364193759, 3.88399987889765925427574582877, 4.77099426953307142103750450811, 5.14313241598963574828292593702, 5.65820857910524980839070467980, 6.92871663068856734968628256461, 7.47623910592486247494257868690