| L(s) = 1 | − 2.56·2-s + 1.05·3-s + 4.56·4-s + 0.0841·5-s − 2.70·6-s + 2.43·7-s − 6.58·8-s − 1.88·9-s − 0.215·10-s − 0.162·11-s + 4.81·12-s + 13-s − 6.23·14-s + 0.0886·15-s + 7.74·16-s + 5.05·17-s + 4.84·18-s − 8.09·19-s + 0.384·20-s + 2.56·21-s + 0.415·22-s − 2.96·23-s − 6.94·24-s − 4.99·25-s − 2.56·26-s − 5.15·27-s + 11.1·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.608·3-s + 2.28·4-s + 0.0376·5-s − 1.10·6-s + 0.919·7-s − 2.32·8-s − 0.629·9-s − 0.0681·10-s − 0.0488·11-s + 1.39·12-s + 0.277·13-s − 1.66·14-s + 0.0228·15-s + 1.93·16-s + 1.22·17-s + 1.14·18-s − 1.85·19-s + 0.0859·20-s + 0.559·21-s + 0.0886·22-s − 0.617·23-s − 1.41·24-s − 0.998·25-s − 0.502·26-s − 0.991·27-s + 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 - 0.0841T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 0.162T + 11T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2.50T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 0.993T + 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 5.16T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−7.85880149245460265254020295268, −7.23486652377418070963641821562, −6.18778739370299361757908565980, −5.84760804286931396593480066659, −4.60848982556967776018046256897, −3.63751023289109080999015652666, −2.61009331678232267182381148079, −2.05257514188543722738498541960, −1.21298147818664327792983187640, 0,
1.21298147818664327792983187640, 2.05257514188543722738498541960, 2.61009331678232267182381148079, 3.63751023289109080999015652666, 4.60848982556967776018046256897, 5.84760804286931396593480066659, 6.18778739370299361757908565980, 7.23486652377418070963641821562, 7.85880149245460265254020295268
