| L(s) = 1 | − 1.95·3-s + 2.28·7-s + 0.840·9-s − 1.12·11-s − 5.95·13-s − 5.80·17-s + 4.08·19-s − 4.47·21-s − 23-s + 4.23·27-s − 0.408·29-s + 3.19·31-s + 2.20·33-s + 9.80·37-s + 11.6·39-s + 6.27·41-s + 7.75·43-s − 6.40·47-s − 1.78·49-s + 11.3·51-s + 6.73·53-s − 8.00·57-s + 4.75·59-s − 6.33·61-s + 1.91·63-s + 0.283·67-s + 1.95·69-s + ⋯ |
| L(s) = 1 | − 1.13·3-s + 0.863·7-s + 0.280·9-s − 0.338·11-s − 1.65·13-s − 1.40·17-s + 0.936·19-s − 0.976·21-s − 0.208·23-s + 0.814·27-s − 0.0758·29-s + 0.573·31-s + 0.383·33-s + 1.61·37-s + 1.87·39-s + 0.979·41-s + 1.18·43-s − 0.933·47-s − 0.254·49-s + 1.59·51-s + 0.925·53-s − 1.06·57-s + 0.619·59-s − 0.810·61-s + 0.241·63-s + 0.0346·67-s + 0.235·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 29 | \( 1 + 0.408T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 - 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 0.283T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.38006962488508556594337104280, −6.65135872597472857217773789383, −5.91116606533560655748299718116, −5.27246857602298186327747397452, −4.69640188196280382955325195221, −4.25795478494188673316527509724, −2.80282361265818399858729410706, −2.24442570192389634416284308539, −1.00363082298408494622837727823, 0,
1.00363082298408494622837727823, 2.24442570192389634416284308539, 2.80282361265818399858729410706, 4.25795478494188673316527509724, 4.69640188196280382955325195221, 5.27246857602298186327747397452, 5.91116606533560655748299718116, 6.65135872597472857217773789383, 7.38006962488508556594337104280
