| L(s) = 1 | − 5.30·2-s + 20.1·4-s − 13.2·5-s − 7·7-s − 64.2·8-s + 70.1·10-s + 33.2·11-s − 78.9·13-s + 37.1·14-s + 179.·16-s − 61.6·17-s + 26.3·19-s − 266.·20-s − 176.·22-s + 23·23-s + 49.8·25-s + 418.·26-s − 140.·28-s − 171.·29-s + 275.·31-s − 439.·32-s + 327.·34-s + 92.5·35-s − 381.·37-s − 139.·38-s + 850.·40-s − 344.·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.51·4-s − 1.18·5-s − 0.377·7-s − 2.84·8-s + 2.21·10-s + 0.912·11-s − 1.68·13-s + 0.708·14-s + 2.81·16-s − 0.879·17-s + 0.317·19-s − 2.97·20-s − 1.71·22-s + 0.208·23-s + 0.398·25-s + 3.15·26-s − 0.950·28-s − 1.09·29-s + 1.59·31-s − 2.42·32-s + 1.64·34-s + 0.447·35-s − 1.69·37-s − 0.595·38-s + 3.36·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1297264711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1297264711\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 5.30T + 8T^{2} \) |
| 5 | \( 1 + 13.2T + 125T^{2} \) |
| 11 | \( 1 - 33.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 349.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 261.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 244.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 29.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 798.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 329.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 631.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 896.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 58.9T + 9.12e5T^{2} \) |
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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
−9.082395734625343591133683464997, −8.453772650492492727279959535453, −7.67405676153715870640885400564, −6.98829875747167735934646425302, −6.55850611471607121456786289976, −5.02610059626740261612212209083, −3.74960109783267865228242261281, −2.71373127570266055184222306188, −1.59371357856264685690127137748, −0.23277107633089784285544479750,
0.23277107633089784285544479750, 1.59371357856264685690127137748, 2.71373127570266055184222306188, 3.74960109783267865228242261281, 5.02610059626740261612212209083, 6.55850611471607121456786289976, 6.98829875747167735934646425302, 7.67405676153715870640885400564, 8.453772650492492727279959535453, 9.082395734625343591133683464997
