| L(s) = 1 | + 2.93·3-s + 3.50·5-s + 3.59·7-s + 5.62·9-s + 2.69·11-s − 2.54·13-s + 10.2·15-s + 2.23·17-s − 7.67·19-s + 10.5·21-s − 1.01·23-s + 7.28·25-s + 7.70·27-s + 5.38·29-s − 2.95·31-s + 7.91·33-s + 12.6·35-s − 2.73·37-s − 7.47·39-s + 3.58·41-s − 11.3·43-s + 19.7·45-s − 1.21·47-s + 5.93·49-s + 6.55·51-s − 11.6·53-s + 9.44·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s + 1.56·5-s + 1.35·7-s + 1.87·9-s + 0.812·11-s − 0.705·13-s + 2.65·15-s + 0.541·17-s − 1.76·19-s + 2.30·21-s − 0.211·23-s + 1.45·25-s + 1.48·27-s + 0.999·29-s − 0.530·31-s + 1.37·33-s + 2.13·35-s − 0.448·37-s − 1.19·39-s + 0.560·41-s − 1.73·43-s + 2.93·45-s − 0.176·47-s + 0.847·49-s + 0.917·51-s − 1.60·53-s + 1.27·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.258141629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.258141629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 3.58T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 + 1.42T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 7.93T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−8.172894512812615024879192696229, −7.62502588808510787723607990318, −6.69329724529006354600094259888, −6.08415287909074536551061740564, −4.95612345354179982240830576025, −4.51097320606058470861630191013, −3.47914606601648521898144606322, −2.55099760202724425881369513936, −1.86101859866089130792548650346, −1.51505297715019024927193510779,
1.51505297715019024927193510779, 1.86101859866089130792548650346, 2.55099760202724425881369513936, 3.47914606601648521898144606322, 4.51097320606058470861630191013, 4.95612345354179982240830576025, 6.08415287909074536551061740564, 6.69329724529006354600094259888, 7.62502588808510787723607990318, 8.172894512812615024879192696229
