| L(s) = 1 | + 2.63·2-s + 1.98·3-s + 4.94·4-s + 5-s + 5.24·6-s − 3.28·7-s + 7.77·8-s + 0.957·9-s + 2.63·10-s − 3.22·11-s + 9.84·12-s − 8.65·14-s + 1.98·15-s + 10.5·16-s − 4.25·17-s + 2.52·18-s − 2.87·19-s + 4.94·20-s − 6.52·21-s − 8.51·22-s + 6.09·23-s + 15.4·24-s + 25-s − 4.06·27-s − 16.2·28-s + 5.77·29-s + 5.24·30-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 1.14·3-s + 2.47·4-s + 0.447·5-s + 2.14·6-s − 1.24·7-s + 2.74·8-s + 0.319·9-s + 0.833·10-s − 0.973·11-s + 2.84·12-s − 2.31·14-s + 0.513·15-s + 2.64·16-s − 1.03·17-s + 0.595·18-s − 0.659·19-s + 1.10·20-s − 1.42·21-s − 1.81·22-s + 1.27·23-s + 3.15·24-s + 0.200·25-s − 0.781·27-s − 3.06·28-s + 1.07·29-s + 0.957·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.950882671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.950882671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 - 1.98T + 3T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 + 0.835T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 2.18T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 + 0.144T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 7.41T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−10.42545029678338512196375500161, −9.350334929396781131633889896657, −8.451171030646510962991812977522, −7.23671863982096264869053909768, −6.57442652165715219414876639072, −5.69981998660827483901314900837, −4.71162038832045911145369612094, −3.64765533810816480567184042038, −2.81942916635934601420539355929, −2.30117736770700978261878788885,
2.30117736770700978261878788885, 2.81942916635934601420539355929, 3.64765533810816480567184042038, 4.71162038832045911145369612094, 5.69981998660827483901314900837, 6.57442652165715219414876639072, 7.23671863982096264869053909768, 8.451171030646510962991812977522, 9.350334929396781131633889896657, 10.42545029678338512196375500161
