| L(s) = 1 | − 2-s + 4-s − 3.90·5-s + 2.61·7-s − 8-s + 3.90·10-s − 4.06·11-s + 3.58·13-s − 2.61·14-s + 16-s + 7.39·17-s − 3.90·20-s + 4.06·22-s − 3.11·23-s + 10.2·25-s − 3.58·26-s + 2.61·28-s − 2.95·29-s − 5.01·31-s − 32-s − 7.39·34-s − 10.2·35-s − 5.60·37-s + 3.90·40-s + 10.5·41-s − 2.33·43-s − 4.06·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.74·5-s + 0.989·7-s − 0.353·8-s + 1.23·10-s − 1.22·11-s + 0.994·13-s − 0.699·14-s + 0.250·16-s + 1.79·17-s − 0.872·20-s + 0.865·22-s − 0.649·23-s + 2.04·25-s − 0.703·26-s + 0.494·28-s − 0.547·29-s − 0.901·31-s − 0.176·32-s − 1.26·34-s − 1.72·35-s − 0.922·37-s + 0.616·40-s + 1.64·41-s − 0.356·43-s − 0.612·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9260245730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9260245730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 - 7.39T + 17T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 1.56T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−7.964445767256968873910776550402, −7.63153425453676837801635987893, −7.04644982228322331516459205699, −5.75458542542673099603629981209, −5.29135448763598536263116987742, −4.21124639056291315882540361898, −3.62897298022401453426644455775, −2.79710100588416224143365058208, −1.56017671354180615310785790715, −0.57809823308015889565866838940,
0.57809823308015889565866838940, 1.56017671354180615310785790715, 2.79710100588416224143365058208, 3.62897298022401453426644455775, 4.21124639056291315882540361898, 5.29135448763598536263116987742, 5.75458542542673099603629981209, 7.04644982228322331516459205699, 7.63153425453676837801635987893, 7.964445767256968873910776550402
