| L(s) = 1 | − 2.09·2-s − 1.91·3-s + 2.39·4-s + 4.00·6-s − 3.06·7-s − 0.817·8-s + 0.659·9-s − 4.57·12-s + 3.04·13-s + 6.42·14-s − 3.06·16-s − 0.463·17-s − 1.38·18-s + 7.89·19-s + 5.86·21-s + 1.39·23-s + 1.56·24-s − 6.39·26-s + 4.47·27-s − 7.33·28-s − 3.72·29-s − 10.4·31-s + 8.06·32-s + 0.971·34-s + 1.57·36-s − 1.84·37-s − 16.5·38-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 1.10·3-s + 1.19·4-s + 1.63·6-s − 1.15·7-s − 0.289·8-s + 0.219·9-s − 1.31·12-s + 0.845·13-s + 1.71·14-s − 0.766·16-s − 0.112·17-s − 0.325·18-s + 1.81·19-s + 1.28·21-s + 0.289·23-s + 0.319·24-s − 1.25·26-s + 0.861·27-s − 1.38·28-s − 0.691·29-s − 1.88·31-s + 1.42·32-s + 0.166·34-s + 0.262·36-s − 0.303·37-s − 2.68·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3270237816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3270237816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 0.463T + 17T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 + 1.31T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 + 4.18T + 53T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 8.91T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−9.013567012343228135378672881212, −8.003914288528850591682903305273, −7.19930713027911286088641035046, −6.68022345540885886701494056052, −5.81691590554486114121313519107, −5.23802110696146232620674774433, −3.86939688507364384276563025989, −2.92120736635799387058612030709, −1.48185850711831650133751945107, −0.48608304695941983178505107545,
0.48608304695941983178505107545, 1.48185850711831650133751945107, 2.92120736635799387058612030709, 3.86939688507364384276563025989, 5.23802110696146232620674774433, 5.81691590554486114121313519107, 6.68022345540885886701494056052, 7.19930713027911286088641035046, 8.003914288528850591682903305273, 9.013567012343228135378672881212
