| L(s) = 1 | − 2-s + 4-s + 4.31·5-s − 0.737·7-s − 8-s − 4.31·10-s + 0.384·11-s − 4.56·13-s + 0.737·14-s + 16-s + 2.02·17-s − 3.92·19-s + 4.31·20-s − 0.384·22-s + 0.886·23-s + 13.5·25-s + 4.56·26-s − 0.737·28-s + 2.08·29-s − 8.67·31-s − 32-s − 2.02·34-s − 3.17·35-s + 9.26·37-s + 3.92·38-s − 4.31·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.92·5-s − 0.278·7-s − 0.353·8-s − 1.36·10-s + 0.115·11-s − 1.26·13-s + 0.197·14-s + 0.250·16-s + 0.490·17-s − 0.900·19-s + 0.964·20-s − 0.0820·22-s + 0.184·23-s + 2.71·25-s + 0.895·26-s − 0.139·28-s + 0.386·29-s − 1.55·31-s − 0.176·32-s − 0.346·34-s − 0.537·35-s + 1.52·37-s + 0.636·38-s − 0.681·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990526774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990526774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 + 0.737T + 7T^{2} \) |
| 11 | \( 1 - 0.384T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 - 0.886T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 8.67T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.744T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 + 1.42T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.64T + 89T^{2} \) |
| 97 | \( 1 - 1.97T + 97T^{2} \) |
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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
−7.83502157578627272976090637385, −7.06647421864604252841391364458, −6.48458378767846233357473239672, −5.84932337315999289544467121610, −5.28538575017979088008932069617, −4.43408944852573105538754595043, −3.13847770315519822952511839612, −2.36319956041454136774543001902, −1.88341995195798520643504620630, −0.76927225156894538180239553773,
0.76927225156894538180239553773, 1.88341995195798520643504620630, 2.36319956041454136774543001902, 3.13847770315519822952511839612, 4.43408944852573105538754595043, 5.28538575017979088008932069617, 5.84932337315999289544467121610, 6.48458378767846233357473239672, 7.06647421864604252841391364458, 7.83502157578627272976090637385
