| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s − 13-s + 15-s + 17-s + 4·19-s + 25-s − 27-s + 2·29-s + 4·33-s − 6·37-s + 39-s − 6·41-s + 4·43-s − 45-s − 7·49-s − 51-s − 6·53-s + 4·55-s − 4·57-s − 4·59-s − 6·61-s + 65-s − 12·67-s − 16·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.124·65-s − 1.46·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.29019432401515, −13.07798214631531, −12.30338879246155, −12.19984790116066, −11.59294956762343, −11.23013552777289, −10.51966562207052, −10.39213367003526, −9.861085281602112, −9.254001464123825, −8.771546632629244, −8.154235760905125, −7.626730350321390, −7.453042684472257, −6.787074524230378, −6.263112148680510, −5.630922829594392, −5.268698968772926, −4.727172709711678, −4.363022506838399, −3.495108324068941, −3.067246547794123, −2.572582198453163, −1.641503458001282, −1.179010314759953, 0, 0,
1.179010314759953, 1.641503458001282, 2.572582198453163, 3.067246547794123, 3.495108324068941, 4.363022506838399, 4.727172709711678, 5.268698968772926, 5.630922829594392, 6.263112148680510, 6.787074524230378, 7.453042684472257, 7.626730350321390, 8.154235760905125, 8.771546632629244, 9.254001464123825, 9.861085281602112, 10.39213367003526, 10.51966562207052, 11.23013552777289, 11.59294956762343, 12.19984790116066, 12.30338879246155, 13.07798214631531, 13.29019432401515
