| L(s) = 1 | − 2-s + 4-s + 1.30·5-s + 7-s − 8-s − 1.30·10-s + 4.30·11-s − 0.605·13-s − 14-s + 16-s + 19-s + 1.30·20-s − 4.30·22-s − 2.60·23-s − 3.30·25-s + 0.605·26-s + 28-s + 9.90·29-s + 7.21·31-s − 32-s + 1.30·35-s + 5.90·37-s − 38-s − 1.30·40-s + 1.30·41-s − 2.69·43-s + 4.30·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.582·5-s + 0.377·7-s − 0.353·8-s − 0.411·10-s + 1.29·11-s − 0.167·13-s − 0.267·14-s + 0.250·16-s + 0.229·19-s + 0.291·20-s − 0.917·22-s − 0.543·23-s − 0.660·25-s + 0.118·26-s + 0.188·28-s + 1.83·29-s + 1.29·31-s − 0.176·32-s + 0.220·35-s + 0.971·37-s − 0.162·38-s − 0.205·40-s + 0.203·41-s − 0.411·43-s + 0.648·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682800972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682800972\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−8.974173038857062072842449375046, −8.278492448305657074646989458852, −7.59892525676571574826897773990, −6.41419493288544355415798987996, −6.30542017675326017872311343331, −5.02830820173614971157011356609, −4.15361547119421797286383868612, −2.97692665940707432912947827403, −1.92281314483725312982576680740, −0.983447341182025989187843292750,
0.983447341182025989187843292750, 1.92281314483725312982576680740, 2.97692665940707432912947827403, 4.15361547119421797286383868612, 5.02830820173614971157011356609, 6.30542017675326017872311343331, 6.41419493288544355415798987996, 7.59892525676571574826897773990, 8.278492448305657074646989458852, 8.974173038857062072842449375046
