| L(s) = 1 | + 3.31·3-s + 1.75·5-s + 2.38·7-s + 8.00·9-s − 3.93·11-s − 0.976·13-s + 5.83·15-s − 2.10·17-s + 6.21·19-s + 7.89·21-s + 6.40·23-s − 1.90·25-s + 16.5·27-s + 0.867·29-s − 10.0·31-s − 13.0·33-s + 4.18·35-s + 0.819·37-s − 3.24·39-s + 4.02·41-s + 6.72·43-s + 14.0·45-s + 8.59·47-s − 1.32·49-s − 6.99·51-s + 2.10·53-s − 6.91·55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 0.786·5-s + 0.900·7-s + 2.66·9-s − 1.18·11-s − 0.270·13-s + 1.50·15-s − 0.511·17-s + 1.42·19-s + 1.72·21-s + 1.33·23-s − 0.381·25-s + 3.19·27-s + 0.161·29-s − 1.79·31-s − 2.26·33-s + 0.708·35-s + 0.134·37-s − 0.518·39-s + 0.628·41-s + 1.02·43-s + 2.09·45-s + 1.25·47-s − 0.189·49-s − 0.979·51-s + 0.289·53-s − 0.932·55-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
| \(L(1)\) |
\(\approx\) |
\(5.543841732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.543841732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + 0.976T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 6.21T + 19T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 - 0.867T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.819T + 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 8.59T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 + 7.00T + 73T^{2} \) |
| 79 | \( 1 + 0.747T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 - 0.430T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−7.996717940916762129546229360167, −7.49419635242817362217119003884, −7.17888740123052145320633647801, −5.81986284228883724556836269101, −5.07152819226287950919730799823, −4.40058009432006782331777422865, −3.37690609289512586287219698122, −2.68051782352774922817016673460, −2.08432760779280943114439613017, −1.26066097150147699733231382219,
1.26066097150147699733231382219, 2.08432760779280943114439613017, 2.68051782352774922817016673460, 3.37690609289512586287219698122, 4.40058009432006782331777422865, 5.07152819226287950919730799823, 5.81986284228883724556836269101, 7.17888740123052145320633647801, 7.49419635242817362217119003884, 7.996717940916762129546229360167
