| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 6·11-s − 13-s − 15-s + 17-s − 19-s − 21-s + 4·23-s + 25-s + 27-s − 8·29-s + 3·31-s − 6·33-s + 35-s − 6·37-s − 39-s − 3·41-s + 3·43-s − 45-s + 7·47-s + 49-s + 51-s + 53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 1.04·33-s + 0.169·35-s − 0.986·37-s − 0.160·39-s − 0.468·41-s + 0.457·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s + 0.140·51-s + 0.137·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 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
−12.85100364435503, −12.67526199817146, −12.18468646450864, −11.50145936604756, −11.18562254154310, −10.50386382299369, −10.28989422408245, −9.914771534811667, −9.203277420277571, −8.784711905496993, −8.513003936224208, −7.721507296292751, −7.564105480227815, −7.191516343277937, −6.624903284334271, −5.814506128994979, −5.545749663257386, −4.988254459180309, −4.420460023353899, −3.985005049560959, −3.227482723948197, −2.922115775030706, −2.506409506306314, −1.787960582009011, −1.138467712007955, 0, 0,
1.138467712007955, 1.787960582009011, 2.506409506306314, 2.922115775030706, 3.227482723948197, 3.985005049560959, 4.420460023353899, 4.988254459180309, 5.545749663257386, 5.814506128994979, 6.624903284334271, 7.191516343277937, 7.564105480227815, 7.721507296292751, 8.513003936224208, 8.784711905496993, 9.203277420277571, 9.914771534811667, 10.28989422408245, 10.50386382299369, 11.18562254154310, 11.50145936604756, 12.18468646450864, 12.67526199817146, 12.85100364435503
