| L(s) = 1 | + 2.65·3-s + 3.67·5-s − 1.77·7-s + 4.06·9-s − 0.583·11-s − 2.83·13-s + 9.76·15-s + 2.11·17-s − 19-s − 4.72·21-s + 2.55·23-s + 8.48·25-s + 2.84·27-s + 8.08·29-s − 4.01·31-s − 1.55·33-s − 6.52·35-s + 9.95·37-s − 7.54·39-s + 9.89·41-s + 0.884·43-s + 14.9·45-s − 4.10·47-s − 3.84·49-s + 5.62·51-s + 7.29·53-s − 2.14·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 1.64·5-s − 0.671·7-s + 1.35·9-s − 0.176·11-s − 0.786·13-s + 2.52·15-s + 0.512·17-s − 0.229·19-s − 1.03·21-s + 0.533·23-s + 1.69·25-s + 0.546·27-s + 1.50·29-s − 0.720·31-s − 0.270·33-s − 1.10·35-s + 1.63·37-s − 1.20·39-s + 1.54·41-s + 0.134·43-s + 2.22·45-s − 0.598·47-s − 0.549·49-s + 0.787·51-s + 1.00·53-s − 0.289·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.705597377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.705597377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 0.583T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 2.11T + 17T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 - 9.95T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 0.884T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 83 | \( 1 - 8.07T + 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.125438161671233798622475472649, −7.49173555720314594520354090534, −6.67027774134817732463271843637, −6.05005586178170681900998625200, −5.22886721136202482885212147892, −4.36988332346696615527860870531, −3.30719140887082151485083304684, −2.61748339617060825910771916033, −2.24295185141327027181608880690, −1.11178690530763329116640388881,
1.11178690530763329116640388881, 2.24295185141327027181608880690, 2.61748339617060825910771916033, 3.30719140887082151485083304684, 4.36988332346696615527860870531, 5.22886721136202482885212147892, 6.05005586178170681900998625200, 6.67027774134817732463271843637, 7.49173555720314594520354090534, 8.125438161671233798622475472649
