| L(s) = 1 | − 1.33·2-s + 1.40·3-s − 0.214·4-s − 1.87·6-s − 0.850·7-s + 2.95·8-s − 1.02·9-s − 2.89·11-s − 0.301·12-s − 6.34·13-s + 1.13·14-s − 3.52·16-s − 4.06·17-s + 1.36·18-s − 4.69·19-s − 1.19·21-s + 3.86·22-s − 0.545·23-s + 4.15·24-s + 8.47·26-s − 5.65·27-s + 0.182·28-s − 9.52·29-s + 7.52·31-s − 1.20·32-s − 4.06·33-s + 5.42·34-s + ⋯ |
| L(s) = 1 | − 0.944·2-s + 0.811·3-s − 0.107·4-s − 0.766·6-s − 0.321·7-s + 1.04·8-s − 0.341·9-s − 0.871·11-s − 0.0870·12-s − 1.75·13-s + 0.303·14-s − 0.881·16-s − 0.985·17-s + 0.322·18-s − 1.07·19-s − 0.260·21-s + 0.823·22-s − 0.113·23-s + 0.848·24-s + 1.66·26-s − 1.08·27-s + 0.0344·28-s − 1.76·29-s + 1.35·31-s − 0.213·32-s − 0.707·33-s + 0.930·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3663547453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3663547453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 0.850T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 0.545T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 0.993T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 - 9.01T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.120954813370835913713735678808, −7.60961388365900790110758077310, −7.07180912165943392700016533176, −6.04295581180750434523248997558, −5.06557844230730156818262403711, −4.47125291521868878609275244373, −3.51668222190672849408507652457, −2.36762593557760513309580515762, −2.13823365709983622618728456160, −0.32625444619364523815385812912,
0.32625444619364523815385812912, 2.13823365709983622618728456160, 2.36762593557760513309580515762, 3.51668222190672849408507652457, 4.47125291521868878609275244373, 5.06557844230730156818262403711, 6.04295581180750434523248997558, 7.07180912165943392700016533176, 7.60961388365900790110758077310, 8.120954813370835913713735678808
