| L(s) = 1 | + 2.71·5-s − 2.29·7-s − 2.71·11-s − 0.130·13-s − 5.29·17-s + 2.49·19-s − 2.05·23-s + 2.39·25-s + 5.85·29-s + 0.277·31-s − 6.24·35-s + 9.10·37-s + 8.78·41-s − 7.14·43-s + 1.16·47-s − 1.72·49-s + 1.51·53-s − 7.39·55-s − 6.58·59-s − 3.25·61-s − 0.356·65-s − 11.6·67-s + 10.5·71-s − 13.9·73-s + 6.24·77-s − 1.32·79-s − 15.3·83-s + ⋯ |
| L(s) = 1 | + 1.21·5-s − 0.867·7-s − 0.819·11-s − 0.0363·13-s − 1.28·17-s + 0.572·19-s − 0.428·23-s + 0.478·25-s + 1.08·29-s + 0.0498·31-s − 1.05·35-s + 1.49·37-s + 1.37·41-s − 1.09·43-s + 0.170·47-s − 0.246·49-s + 0.207·53-s − 0.996·55-s − 0.857·59-s − 0.416·61-s − 0.0441·65-s − 1.42·67-s + 1.25·71-s − 1.63·73-s + 0.711·77-s − 0.149·79-s − 1.68·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 + 0.130T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 0.277T + 31T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 - 1.51T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 1.32T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 0.650T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.70911413699567354592089329451, −6.86155190655434222830256145412, −6.21049023974391089036965375054, −5.77608266209380181658483211597, −4.88616510952164231173960217006, −4.13792123369276741832255580400, −2.87017305111885516520118325037, −2.53238077682676441617042125081, −1.40926218852153424849170099262, 0,
1.40926218852153424849170099262, 2.53238077682676441617042125081, 2.87017305111885516520118325037, 4.13792123369276741832255580400, 4.88616510952164231173960217006, 5.77608266209380181658483211597, 6.21049023974391089036965375054, 6.86155190655434222830256145412, 7.70911413699567354592089329451
