| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 11-s + 13-s + 14-s − 16-s − 5·19-s + 22-s − 23-s + 26-s − 28-s + 5·29-s − 2·31-s + 5·32-s − 4·37-s − 5·38-s + 5·41-s − 9·43-s − 44-s − 46-s + 6·47-s − 6·49-s − 52-s − 2·53-s − 3·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 0.301·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s − 1.14·19-s + 0.213·22-s − 0.208·23-s + 0.196·26-s − 0.188·28-s + 0.928·29-s − 0.359·31-s + 0.883·32-s − 0.657·37-s − 0.811·38-s + 0.780·41-s − 1.37·43-s − 0.150·44-s − 0.147·46-s + 0.875·47-s − 6/7·49-s − 0.138·52-s − 0.274·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.078872118289728758345274607878, −6.91527039034194976795635691340, −6.30590621067405743132248971334, −5.58064116684284011323432825785, −4.80035180751949106365438436353, −4.22396979433871816607697639941, −3.49670554018888528164512843131, −2.56726561832303611472281478831, −1.41553409351047109970547139994, 0,
1.41553409351047109970547139994, 2.56726561832303611472281478831, 3.49670554018888528164512843131, 4.22396979433871816607697639941, 4.80035180751949106365438436353, 5.58064116684284011323432825785, 6.30590621067405743132248971334, 6.91527039034194976795635691340, 8.078872118289728758345274607878
