| L(s) = 1 | + 0.949·3-s − 3.85i·7-s − 2.09·9-s − 0.589i·11-s + (−1.38 − 3.32i)13-s + 3.66·17-s + 5.94i·19-s − 3.66i·21-s − 3.51·23-s − 4.83·27-s − 5.33·29-s − 8.71i·31-s − 0.559i·33-s − 1.85i·37-s + (−1.31 − 3.15i)39-s + ⋯ |
| L(s) = 1 | + 0.547·3-s − 1.45i·7-s − 0.699·9-s − 0.177i·11-s + (−0.384 − 0.922i)13-s + 0.887·17-s + 1.36i·19-s − 0.798i·21-s − 0.733·23-s − 0.931·27-s − 0.991·29-s − 1.56i·31-s − 0.0974i·33-s − 0.305i·37-s + (−0.210 − 0.505i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9603929501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9603929501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1.38 + 3.32i)T \) |
| good | 3 | \( 1 - 0.949T + 3T^{2} \) |
| 7 | \( 1 + 3.85iT - 7T^{2} \) |
| 11 | \( 1 + 0.589iT - 11T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 - 5.94iT - 19T^{2} \) |
| 23 | \( 1 + 3.51T + 23T^{2} \) |
| 29 | \( 1 + 5.33T + 29T^{2} \) |
| 31 | \( 1 + 8.71iT - 31T^{2} \) |
| 37 | \( 1 + 1.85iT - 37T^{2} \) |
| 41 | \( 1 - 4.63iT - 41T^{2} \) |
| 43 | \( 1 - 6.30T + 43T^{2} \) |
| 47 | \( 1 - 3.85iT - 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 4.09iT - 67T^{2} \) |
| 71 | \( 1 - 5.34iT - 71T^{2} \) |
| 73 | \( 1 + 6.09iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 8.77iT - 83T^{2} \) |
| 89 | \( 1 + 0.413iT - 89T^{2} \) |
| 97 | \( 1 + 8.45iT - 97T^{2} \) |
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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
−8.198465932858962198663237580035, −7.87242420321189745966847755143, −7.36376985018316724279367243017, −6.06806236203967602069074025461, −5.61119758956842401658096704407, −4.33454757795301282537386655016, −3.63844329166586592136012323717, −2.90551638609907472071121191613, −1.60015709880850443246246656042, −0.27186865707895990728374793173,
1.78857750638616321217703597728, 2.61122089702861409736209211725, 3.30601151167861648601106398165, 4.53830469092636401313926339956, 5.39685056421901802658329815988, 6.01181537977732132483028712669, 6.98904893922557594997691441322, 7.79461102340777390680649290739, 8.662816841802284282220294918761, 9.108103923735082830078603907463
