L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 4·19-s − 25-s + 27-s − 10·29-s − 8·31-s − 4·33-s − 2·37-s + 2·39-s − 10·41-s − 12·43-s − 2·45-s − 7·49-s − 6·53-s + 8·55-s − 4·57-s − 12·59-s − 10·61-s − 4·65-s + 12·67-s − 10·73-s − 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s − 1.56·41-s − 1.82·43-s − 0.298·45-s − 49-s − 0.824·53-s + 1.07·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.17·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.98706748455872, −14.55422575064723, −13.82512236985294, −13.27402700416461, −12.98263546337090, −12.52360873409305, −11.79089672241186, −11.33479094708476, −10.75060334355783, −10.47031453701398, −9.656119004900982, −9.221118082372303, −8.520878410234351, −8.128033060420344, −7.726591488849617, −7.191694135733382, −6.577129390178241, −5.869535976188814, −5.192655171463674, −4.696462894525261, −3.905649836923843, −3.457248351640454, −3.009774989190016, −1.923390589108132, −1.673243742138415, 0, 0,
1.673243742138415, 1.923390589108132, 3.009774989190016, 3.457248351640454, 3.905649836923843, 4.696462894525261, 5.192655171463674, 5.869535976188814, 6.577129390178241, 7.191694135733382, 7.726591488849617, 8.128033060420344, 8.520878410234351, 9.221118082372303, 9.656119004900982, 10.47031453701398, 10.75060334355783, 11.33479094708476, 11.79089672241186, 12.52360873409305, 12.98263546337090, 13.27402700416461, 13.82512236985294, 14.55422575064723, 14.98706748455872