| L(s) = 1 | + 2.01·3-s − 1.57·5-s + 1.04·9-s − 4.02·11-s + 5.31·13-s − 3.16·15-s − 17-s + 2.04·19-s − 2.80·23-s − 2.52·25-s − 3.93·27-s − 6.93·29-s + 8.06·31-s − 8.08·33-s + 4.04·37-s + 10.6·39-s + 4.49·41-s − 2.57·43-s − 1.63·45-s − 1.82·47-s − 2.01·51-s + 2.98·53-s + 6.33·55-s + 4.11·57-s − 1.54·59-s − 11.3·61-s − 8.37·65-s + ⋯ |
| L(s) = 1 | + 1.16·3-s − 0.703·5-s + 0.346·9-s − 1.21·11-s + 1.47·13-s − 0.816·15-s − 0.242·17-s + 0.469·19-s − 0.584·23-s − 0.504·25-s − 0.758·27-s − 1.28·29-s + 1.44·31-s − 1.40·33-s + 0.665·37-s + 1.71·39-s + 0.701·41-s − 0.393·43-s − 0.244·45-s − 0.266·47-s − 0.281·51-s + 0.409·53-s + 0.853·55-s + 0.545·57-s − 0.201·59-s − 1.45·61-s − 1.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 - 4.04T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 + 2.57T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 - 8.78T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 0.454T + 97T^{2} \) |
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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
−7.81186030471692645714065965679, −7.30326300731467432345970114852, −6.14116091334694066217839693195, −5.63530825353418482610459204089, −4.49426597307691113660709935216, −3.86351095244876942430419773759, −3.15266590457142829522555815215, −2.51097918172718304520595408460, −1.45507769136425137047447284250, 0,
1.45507769136425137047447284250, 2.51097918172718304520595408460, 3.15266590457142829522555815215, 3.86351095244876942430419773759, 4.49426597307691113660709935216, 5.63530825353418482610459204089, 6.14116091334694066217839693195, 7.30326300731467432345970114852, 7.81186030471692645714065965679
