L(s) = 1 | − 2.68·2-s + 5.19·4-s − 7-s − 8.56·8-s + 3.19·11-s + 6.38·13-s + 2.68·14-s + 12.5·16-s − 5.36·17-s − 2.38·19-s − 8.57·22-s + 3.19·23-s − 17.1·26-s − 5.19·28-s + 2.16·29-s + 6·31-s − 16.6·32-s + 14.3·34-s + 3·37-s + 6.39·38-s − 10.7·41-s − 9.38·43-s + 16.6·44-s − 8.57·46-s + 11.7·47-s + 49-s + 33.1·52-s + ⋯ |
L(s) = 1 | − 1.89·2-s + 2.59·4-s − 0.377·7-s − 3.02·8-s + 0.964·11-s + 1.77·13-s + 0.716·14-s + 3.14·16-s − 1.30·17-s − 0.547·19-s − 1.82·22-s + 0.666·23-s − 3.35·26-s − 0.981·28-s + 0.402·29-s + 1.07·31-s − 2.93·32-s + 2.46·34-s + 0.493·37-s + 1.03·38-s − 1.67·41-s − 1.43·43-s + 2.50·44-s − 1.26·46-s + 1.71·47-s + 0.142·49-s + 4.59·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7330294630\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7330294630\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 - 6.38T + 13T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.38T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 9.38T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.140791689210818874293317318519, −8.719817428349082423391649808555, −8.282645142947836076338349666784, −6.94614687982193229259735236660, −6.64410108836379703747592469041, −5.88074145088887858900943359860, −4.15305800816045799105803153411, −3.03754061987304116655603212095, −1.84243654883089816851595569051, −0.820980777020587896750483728234,
0.820980777020587896750483728234, 1.84243654883089816851595569051, 3.03754061987304116655603212095, 4.15305800816045799105803153411, 5.88074145088887858900943359860, 6.64410108836379703747592469041, 6.94614687982193229259735236660, 8.282645142947836076338349666784, 8.719817428349082423391649808555, 9.140791689210818874293317318519