| L(s) = 1 | − 3·3-s − 5·7-s + 9·9-s + 14·11-s − 13-s − 46·17-s + 19·19-s + 15·21-s + 46·23-s − 27·27-s + 14·29-s + 133·31-s − 42·33-s − 258·37-s + 3·39-s + 84·41-s + 167·43-s − 410·47-s − 318·49-s + 138·51-s − 456·53-s − 57·57-s − 194·59-s − 17·61-s − 45·63-s − 653·67-s − 138·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.269·7-s + 1/3·9-s + 0.383·11-s − 0.0213·13-s − 0.656·17-s + 0.229·19-s + 0.155·21-s + 0.417·23-s − 0.192·27-s + 0.0896·29-s + 0.770·31-s − 0.221·33-s − 1.14·37-s + 0.0123·39-s + 0.319·41-s + 0.592·43-s − 1.27·47-s − 0.927·49-s + 0.378·51-s − 1.18·53-s − 0.132·57-s − 0.428·59-s − 0.0356·61-s − 0.0899·63-s − 1.19·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - p T + p^{3} T^{2} \) |
| 23 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 - 133 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 84 T + p^{3} T^{2} \) |
| 43 | \( 1 - 167 T + p^{3} T^{2} \) |
| 47 | \( 1 + 410 T + p^{3} T^{2} \) |
| 53 | \( 1 + 456 T + p^{3} T^{2} \) |
| 59 | \( 1 + 194 T + p^{3} T^{2} \) |
| 61 | \( 1 + 17 T + p^{3} T^{2} \) |
| 67 | \( 1 + 653 T + p^{3} T^{2} \) |
| 71 | \( 1 - 828 T + p^{3} T^{2} \) |
| 73 | \( 1 + 570 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 142 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1104 T + p^{3} T^{2} \) |
| 97 | \( 1 + 841 T + p^{3} T^{2} \) |
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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
−9.853776397969120604618428199620, −9.067081188466859593651617878747, −8.037333894389118986660853080570, −6.92828621815004337233921292544, −6.28256720359774686233792727359, −5.18511376787084567594995246369, −4.24248004324726432064316242329, −2.98955944237088359240704654108, −1.46808929779602186516515225409, 0,
1.46808929779602186516515225409, 2.98955944237088359240704654108, 4.24248004324726432064316242329, 5.18511376787084567594995246369, 6.28256720359774686233792727359, 6.92828621815004337233921292544, 8.037333894389118986660853080570, 9.067081188466859593651617878747, 9.853776397969120604618428199620
