| L(s) = 1 | − 2.41·2-s + 0.585·3-s + 3.82·4-s + 5-s − 1.41·6-s + 3.41·7-s − 4.41·8-s − 2.65·9-s − 2.41·10-s + 5.41·11-s + 2.24·12-s + 2.82·13-s − 8.24·14-s + 0.585·15-s + 2.99·16-s + 6.41·18-s + 2.82·19-s + 3.82·20-s + 2·21-s − 13.0·22-s + 0.585·23-s − 2.58·24-s + 25-s − 6.82·26-s − 3.31·27-s + 13.0·28-s − 0.828·29-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.338·3-s + 1.91·4-s + 0.447·5-s − 0.577·6-s + 1.29·7-s − 1.56·8-s − 0.885·9-s − 0.763·10-s + 1.63·11-s + 0.647·12-s + 0.784·13-s − 2.20·14-s + 0.151·15-s + 0.749·16-s + 1.51·18-s + 0.648·19-s + 0.856·20-s + 0.436·21-s − 2.78·22-s + 0.122·23-s − 0.527·24-s + 0.200·25-s − 1.33·26-s − 0.637·27-s + 2.47·28-s − 0.153·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188952302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188952302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 0.585T + 3T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 0.828T + 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−9.315906738616028568293273386454, −8.683633444625156155550883722069, −8.298616307969767276460219952683, −7.42571342400959106525303714136, −6.51304531279506650927430578470, −5.72495462535729063732934081597, −4.39688208364461144224006477716, −3.03982520375947862474758447268, −1.80769991564838708167181287774, −1.11105768879508331296556409031,
1.11105768879508331296556409031, 1.80769991564838708167181287774, 3.03982520375947862474758447268, 4.39688208364461144224006477716, 5.72495462535729063732934081597, 6.51304531279506650927430578470, 7.42571342400959106525303714136, 8.298616307969767276460219952683, 8.683633444625156155550883722069, 9.315906738616028568293273386454
