| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 6·11-s − 13-s + 14-s + 16-s − 17-s + 19-s + 20-s − 6·22-s + 4·23-s + 25-s − 26-s + 28-s + 8·29-s − 3·31-s + 32-s − 34-s + 35-s − 6·37-s + 38-s + 40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s − 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.48·29-s − 0.538·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s − 0.986·37-s + 0.162·38-s + 0.158·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.63591390386107, −13.23713426321660, −12.72940190973678, −12.36346642610775, −11.86431559330682, −11.22277993809634, −10.63925240062930, −10.57104561120124, −9.922778192152255, −9.399350458865093, −8.541137902596185, −8.454261621062488, −7.552669726964427, −7.329072705639309, −6.772899037646347, −6.021218817679484, −5.618227303457792, −5.111185203895113, −4.703948703252687, −4.253101152028589, −3.200853990142891, −2.962712119766315, −2.340273844668392, −1.766728604018872, −0.9456203676272823, 0,
0.9456203676272823, 1.766728604018872, 2.340273844668392, 2.962712119766315, 3.200853990142891, 4.253101152028589, 4.703948703252687, 5.111185203895113, 5.618227303457792, 6.021218817679484, 6.772899037646347, 7.329072705639309, 7.552669726964427, 8.454261621062488, 8.541137902596185, 9.399350458865093, 9.922778192152255, 10.57104561120124, 10.63925240062930, 11.22277993809634, 11.86431559330682, 12.36346642610775, 12.72940190973678, 13.23713426321660, 13.63591390386107
