| L(s) = 1 | − 18.7·5-s − 19.7·7-s + 5.84·11-s + 33.7·13-s + 33.2·17-s + 102.·19-s + 130.·23-s + 228.·25-s − 111.·29-s − 257.·31-s + 370.·35-s + 424.·37-s − 10.9·41-s + 324.·43-s − 599.·47-s + 45.7·49-s − 106.·53-s − 109.·55-s − 617.·59-s − 325.·61-s − 634.·65-s − 240.·67-s + 123.·71-s − 46.7·73-s − 115.·77-s + 547.·79-s + 1.21e3·83-s + ⋯ |
| L(s) = 1 | − 1.68·5-s − 1.06·7-s + 0.160·11-s + 0.719·13-s + 0.474·17-s + 1.24·19-s + 1.17·23-s + 1.82·25-s − 0.715·29-s − 1.49·31-s + 1.78·35-s + 1.88·37-s − 0.0416·41-s + 1.15·43-s − 1.86·47-s + 0.133·49-s − 0.276·53-s − 0.269·55-s − 1.36·59-s − 0.682·61-s − 1.21·65-s − 0.437·67-s + 0.207·71-s − 0.0749·73-s − 0.170·77-s + 0.779·79-s + 1.60·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 + 19.7T + 343T^{2} \) |
| 11 | \( 1 - 5.84T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 424.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 324.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 617.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 46.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−9.130112607380991072672899215117, −7.990647602553267574005879920482, −7.47629900508117810754807757779, −6.65292131436323617816497652196, −5.63176346981725065057332398146, −4.48109461032719560411517147070, −3.50335149687403109067222759661, −3.10805498415469800051914068335, −1.09469591784970631263785587142, 0,
1.09469591784970631263785587142, 3.10805498415469800051914068335, 3.50335149687403109067222759661, 4.48109461032719560411517147070, 5.63176346981725065057332398146, 6.65292131436323617816497652196, 7.47629900508117810754807757779, 7.990647602553267574005879920482, 9.130112607380991072672899215117
