| L(s) = 1 | − 4.46·5-s + 0.464·13-s + 7.92·17-s + 14.9·25-s − 8.46·29-s + 11.3·37-s − 10·41-s − 7·49-s − 14·53-s − 5.39·61-s − 2.07·65-s − 10.8·73-s − 35.3·85-s − 8.85·89-s + 18·97-s + 2·101-s − 20.3·109-s + 6.85·113-s + ⋯ |
| L(s) = 1 | − 1.99·5-s + 0.128·13-s + 1.92·17-s + 2.98·25-s − 1.57·29-s + 1.87·37-s − 1.56·41-s − 49-s − 1.92·53-s − 0.690·61-s − 0.256·65-s − 1.27·73-s − 3.83·85-s − 0.938·89-s + 1.82·97-s + 0.199·101-s − 1.94·109-s + 0.644·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.199234627009631173491892201221, −7.79515094847959590588506878913, −7.28268689162276652758007993137, −6.25691693908072363262966121520, −5.22724566305105273202413373083, −4.40731150190944285276890829366, −3.56203580692724898330422010749, −3.06769458728378563523590662439, −1.29162222912781718445665208888, 0,
1.29162222912781718445665208888, 3.06769458728378563523590662439, 3.56203580692724898330422010749, 4.40731150190944285276890829366, 5.22724566305105273202413373083, 6.25691693908072363262966121520, 7.28268689162276652758007993137, 7.79515094847959590588506878913, 8.199234627009631173491892201221
