| L(s) = 1 | − 2·4-s − 5-s − 2·7-s + 11-s + 4·16-s + 17-s − 19-s + 2·20-s − 8·23-s + 25-s + 4·28-s − 5·29-s + 2·35-s − 6·37-s − 9·41-s − 11·43-s − 2·44-s + 6·47-s − 3·49-s − 4·53-s − 55-s + 10·59-s − 8·64-s + 3·67-s − 2·68-s + 12·73-s + 2·76-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.755·7-s + 0.301·11-s + 16-s + 0.242·17-s − 0.229·19-s + 0.447·20-s − 1.66·23-s + 1/5·25-s + 0.755·28-s − 0.928·29-s + 0.338·35-s − 0.986·37-s − 1.40·41-s − 1.67·43-s − 0.301·44-s + 0.875·47-s − 3/7·49-s − 0.549·53-s − 0.134·55-s + 1.30·59-s − 64-s + 0.366·67-s − 0.242·68-s + 1.40·73-s + 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.74685062932731, −13.28166904558702, −12.75034031203491, −12.37047024884736, −11.86602369937506, −11.50158776414696, −10.68986021927296, −10.23891126956337, −9.785991017610908, −9.475188073815870, −8.790004333714336, −8.389844525421147, −7.972544905897757, −7.401593556840551, −6.664876976105575, −6.391612516614715, −5.568384543365049, −5.244743900798354, −4.574060254401320, −3.916629387804765, −3.590061893108640, −3.184608145922986, −2.145158993413216, −1.575503080920778, −0.5657210711625913, 0,
0.5657210711625913, 1.575503080920778, 2.145158993413216, 3.184608145922986, 3.590061893108640, 3.916629387804765, 4.574060254401320, 5.244743900798354, 5.568384543365049, 6.391612516614715, 6.664876976105575, 7.401593556840551, 7.972544905897757, 8.389844525421147, 8.790004333714336, 9.475188073815870, 9.785991017610908, 10.23891126956337, 10.68986021927296, 11.50158776414696, 11.86602369937506, 12.37047024884736, 12.75034031203491, 13.28166904558702, 13.74685062932731
