| L(s) = 1 | + 2.28·2-s + 3.21·3-s + 3.20·4-s − 5-s + 7.33·6-s − 2.30·7-s + 2.74·8-s + 7.33·9-s − 2.28·10-s − 1.25·11-s + 10.2·12-s − 5.25·14-s − 3.21·15-s − 0.151·16-s + 2.43·17-s + 16.7·18-s − 0.586·19-s − 3.20·20-s − 7.40·21-s − 2.87·22-s − 8.37·23-s + 8.81·24-s + 25-s + 13.9·27-s − 7.37·28-s − 3.09·29-s − 7.33·30-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.85·3-s + 1.60·4-s − 0.447·5-s + 2.99·6-s − 0.870·7-s + 0.969·8-s + 2.44·9-s − 0.721·10-s − 0.379·11-s + 2.97·12-s − 1.40·14-s − 0.829·15-s − 0.0378·16-s + 0.589·17-s + 3.94·18-s − 0.134·19-s − 0.715·20-s − 1.61·21-s − 0.612·22-s − 1.74·23-s + 1.79·24-s + 0.200·25-s + 2.67·27-s − 1.39·28-s − 0.574·29-s − 1.33·30-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
| \(L(1)\) |
\(\approx\) |
\(5.659157126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.659157126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 + 1.25T + 11T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 0.586T + 19T^{2} \) |
| 23 | \( 1 + 8.37T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 + 0.394T + 31T^{2} \) |
| 37 | \( 1 - 2.01T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 + 1.99T + 67T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 + 0.217T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−10.02092280702675673605793610623, −9.406428929021332048953097404704, −8.293844471669924943573534435201, −7.60364094387031289883966480115, −6.72767127205731183856976202500, −5.68623079822733977697406437845, −4.32662425502737115578291735528, −3.74571126374480532100536887103, −3.00518557583222937630958660051, −2.15380176020096694111641852586,
2.15380176020096694111641852586, 3.00518557583222937630958660051, 3.74571126374480532100536887103, 4.32662425502737115578291735528, 5.68623079822733977697406437845, 6.72767127205731183856976202500, 7.60364094387031289883966480115, 8.293844471669924943573534435201, 9.406428929021332048953097404704, 10.02092280702675673605793610623
