| L(s) = 1 | − 1.33·3-s + 3.94·5-s + 0.220·7-s − 1.20·9-s + 1.87·11-s + 3.17·13-s − 5.28·15-s + 7.01·17-s + 5.00·19-s − 0.295·21-s + 1.70·23-s + 10.5·25-s + 5.63·27-s + 1.30·29-s − 4.27·31-s − 2.50·33-s + 0.871·35-s + 3.57·37-s − 4.25·39-s + 4.53·41-s − 3.45·43-s − 4.77·45-s + 1.75·47-s − 6.95·49-s − 9.38·51-s − 6.18·53-s + 7.38·55-s + ⋯ |
| L(s) = 1 | − 0.772·3-s + 1.76·5-s + 0.0834·7-s − 0.402·9-s + 0.564·11-s + 0.880·13-s − 1.36·15-s + 1.70·17-s + 1.14·19-s − 0.0644·21-s + 0.355·23-s + 2.11·25-s + 1.08·27-s + 0.242·29-s − 0.767·31-s − 0.436·33-s + 0.147·35-s + 0.587·37-s − 0.680·39-s + 0.708·41-s − 0.527·43-s − 0.711·45-s + 0.255·47-s − 0.993·49-s − 1.31·51-s − 0.850·53-s + 0.996·55-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
| \(L(1)\) |
\(\approx\) |
\(2.666385503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.666385503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 - 0.220T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 - 3.47T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 + 6.03T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 0.167T + 97T^{2} \) |
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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
−8.107391090498764424685291646998, −7.20449705328941860853544195278, −6.32720607604378554474612604835, −5.92099128969478197866002588263, −5.43736072142244271583253822574, −4.80840696652427336192047815071, −3.47116527788634055978105022408, −2.81041731168986227285852003688, −1.55467869711872911389801039227, −1.01702705051917776136472364941,
1.01702705051917776136472364941, 1.55467869711872911389801039227, 2.81041731168986227285852003688, 3.47116527788634055978105022408, 4.80840696652427336192047815071, 5.43736072142244271583253822574, 5.92099128969478197866002588263, 6.32720607604378554474612604835, 7.20449705328941860853544195278, 8.107391090498764424685291646998
