| L(s) = 1 | + 0.163·3-s − 2.50·5-s − 0.0804·7-s − 2.97·9-s + 1.12·11-s − 6.34·13-s − 0.409·15-s + 3.88·17-s − 6.17·19-s − 0.0131·21-s − 5.65·23-s + 1.25·25-s − 0.978·27-s + 7.85·29-s + 3.07·31-s + 0.183·33-s + 0.201·35-s − 10.6·37-s − 1.03·39-s + 10.6·41-s − 5.76·43-s + 7.43·45-s − 2.63·47-s − 6.99·49-s + 0.635·51-s + 1.17·53-s − 2.80·55-s + ⋯ |
| L(s) = 1 | + 0.0945·3-s − 1.11·5-s − 0.0303·7-s − 0.991·9-s + 0.338·11-s − 1.75·13-s − 0.105·15-s + 0.941·17-s − 1.41·19-s − 0.00287·21-s − 1.17·23-s + 0.251·25-s − 0.188·27-s + 1.45·29-s + 0.552·31-s + 0.0319·33-s + 0.0340·35-s − 1.74·37-s − 0.166·39-s + 1.65·41-s − 0.879·43-s + 1.10·45-s − 0.384·47-s − 0.999·49-s + 0.0889·51-s + 0.161·53-s − 0.378·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4993837622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4993837622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.163T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 + 0.0804T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 2.63T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 8.49T + 59T^{2} \) |
| 61 | \( 1 + 4.59T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 0.796T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 0.612T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.84919580877045567248102789003, −7.32720984241591456818577114994, −6.43983356805008146779588320335, −5.83769577889580816459453479940, −4.78845574164617224087770019234, −4.41357803284473149462930217792, −3.42283392295777097765047513091, −2.81447591131039721252990634782, −1.87295983419753978885013241519, −0.32796274955144405769035736102,
0.32796274955144405769035736102, 1.87295983419753978885013241519, 2.81447591131039721252990634782, 3.42283392295777097765047513091, 4.41357803284473149462930217792, 4.78845574164617224087770019234, 5.83769577889580816459453479940, 6.43983356805008146779588320335, 7.32720984241591456818577114994, 7.84919580877045567248102789003
