| L(s) = 1 | + 1.41·2-s + 3·7-s − 2.82·8-s + 4.24·11-s + 3·13-s + 4.24·14-s − 4.00·16-s − 2.82·17-s + 19-s + 6·22-s + 7.07·23-s + 4.24·26-s − 4.24·29-s + 2·31-s − 4.00·34-s + 9·37-s + 1.41·38-s − 4.24·41-s + 6·43-s + 10.0·46-s − 2.82·47-s + 2·49-s − 9.89·53-s − 8.48·56-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 1.13·7-s − 0.999·8-s + 1.27·11-s + 0.832·13-s + 1.13·14-s − 1.00·16-s − 0.685·17-s + 0.229·19-s + 1.27·22-s + 1.47·23-s + 0.832·26-s − 0.787·29-s + 0.359·31-s − 0.685·34-s + 1.47·37-s + 0.229·38-s − 0.662·41-s + 0.914·43-s + 1.47·46-s − 0.412·47-s + 0.285·49-s − 1.35·53-s − 1.13·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582783153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582783153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
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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
−10.96359787634588376615553466527, −9.354568816709515276368823300071, −8.918690775052199079847952077310, −7.88733753718221590586782687106, −6.67129509723357145602197202319, −5.88126166141498480956011352016, −4.78750080225639109817325650821, −4.19659819879050368867774418745, −3.07835625551688383837876718067, −1.41968084240141269635968831838,
1.41968084240141269635968831838, 3.07835625551688383837876718067, 4.19659819879050368867774418745, 4.78750080225639109817325650821, 5.88126166141498480956011352016, 6.67129509723357145602197202319, 7.88733753718221590586782687106, 8.918690775052199079847952077310, 9.354568816709515276368823300071, 10.96359787634588376615553466527
