| L(s) = 1 | − 2.54·3-s − 2.18·5-s − 2.46·7-s + 3.48·9-s + 5.27·11-s + 2.99·13-s + 5.55·15-s + 4.41·17-s + 7.60·19-s + 6.28·21-s + 4.58·23-s − 0.241·25-s − 1.24·27-s + 1.08·29-s − 9.31·31-s − 13.4·33-s + 5.38·35-s + 2.04·37-s − 7.63·39-s + 2.65·41-s + 5.35·43-s − 7.61·45-s + 10.9·47-s − 0.909·49-s − 11.2·51-s + 2.86·53-s − 11.5·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.975·5-s − 0.932·7-s + 1.16·9-s + 1.59·11-s + 0.831·13-s + 1.43·15-s + 1.07·17-s + 1.74·19-s + 1.37·21-s + 0.955·23-s − 0.0483·25-s − 0.240·27-s + 0.200·29-s − 1.67·31-s − 2.34·33-s + 0.909·35-s + 0.335·37-s − 1.22·39-s + 0.414·41-s + 0.816·43-s − 1.13·45-s + 1.60·47-s − 0.129·49-s − 1.57·51-s + 0.393·53-s − 1.55·55-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
| \(L(1)\) |
\(\approx\) |
\(1.037030228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037030228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 - 7.60T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 + 9.31T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 + 0.454T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 9.55T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−7.77443616412257075210254209695, −7.23656236982542176384652161412, −6.62568565518670378087371269124, −5.86182879853810348191205086412, −5.47907711621984158919599382214, −4.39865740128774138592540528909, −3.68657362743851587874745190656, −3.18408014596762789390589234138, −1.28051765221278945874727473657, −0.68497827826489033033416226976,
0.68497827826489033033416226976, 1.28051765221278945874727473657, 3.18408014596762789390589234138, 3.68657362743851587874745190656, 4.39865740128774138592540528909, 5.47907711621984158919599382214, 5.86182879853810348191205086412, 6.62568565518670378087371269124, 7.23656236982542176384652161412, 7.77443616412257075210254209695
