| L(s) = 1 | + 1.24·2-s + 1.66·3-s − 0.457·4-s + 2.06·6-s + 4.35·7-s − 3.05·8-s − 0.242·9-s + 0.760·11-s − 0.759·12-s + 3.53·13-s + 5.41·14-s − 2.87·16-s + 17-s − 0.300·18-s + 0.972·19-s + 7.23·21-s + 0.945·22-s − 7.47·23-s − 5.06·24-s + 4.39·26-s − 5.38·27-s − 1.99·28-s − 5.25·29-s + 8.62·31-s + 2.53·32-s + 1.26·33-s + 1.24·34-s + ⋯ |
| L(s) = 1 | + 0.878·2-s + 0.958·3-s − 0.228·4-s + 0.842·6-s + 1.64·7-s − 1.07·8-s − 0.0807·9-s + 0.229·11-s − 0.219·12-s + 0.980·13-s + 1.44·14-s − 0.719·16-s + 0.242·17-s − 0.0708·18-s + 0.223·19-s + 1.57·21-s + 0.201·22-s − 1.55·23-s − 1.03·24-s + 0.861·26-s − 1.03·27-s − 0.376·28-s − 0.976·29-s + 1.54·31-s + 0.447·32-s + 0.219·33-s + 0.213·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.734987937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734987937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 3 | \( 1 - 1.66T + 3T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 - 0.760T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 19 | \( 1 - 0.972T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 3.98T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 1.54T + 53T^{2} \) |
| 59 | \( 1 - 2.66T + 59T^{2} \) |
| 61 | \( 1 + 3.32T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
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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
−11.57865059451670590940931722371, −10.29472769276743630664068390779, −9.082243673707404764298609520585, −8.388877621671779673156243229126, −7.82256939048566820956661436682, −6.18583462385291200260309802812, −5.21395535328633499078677321425, −4.20926726214541565482767402750, −3.31201376684038709815969091525, −1.84188267605493801351100008422,
1.84188267605493801351100008422, 3.31201376684038709815969091525, 4.20926726214541565482767402750, 5.21395535328633499078677321425, 6.18583462385291200260309802812, 7.82256939048566820956661436682, 8.388877621671779673156243229126, 9.082243673707404764298609520585, 10.29472769276743630664068390779, 11.57865059451670590940931722371
