| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s − 2·11-s − 4·13-s + 4·15-s − 2·17-s − 2·21-s + 23-s − 25-s + 4·27-s + 2·29-s + 4·33-s − 2·35-s + 4·37-s + 8·39-s + 6·41-s − 2·43-s − 2·45-s + 4·47-s + 49-s + 4·51-s + 4·55-s − 2·59-s + 10·61-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 1.03·15-s − 0.485·17-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s + 0.657·37-s + 1.28·39-s + 0.937·41-s − 0.304·43-s − 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.539·55-s − 0.260·59-s + 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.91208332511826, −16.36244137166378, −15.73058644092489, −15.40485093080377, −14.57061407906520, −14.23265205943321, −13.16773763213170, −12.78469027864833, −12.01115766788416, −11.67278362934388, −11.25602391923793, −10.50403052934678, −10.14544997134552, −9.186398647822894, −8.536017817456366, −7.675937734443965, −7.440212654826769, −6.559171157434498, −5.903646284859285, −5.149935606091350, −4.686139442532922, −4.039947963096991, −2.963589363880053, −2.179383177492332, −0.8213983700428921, 0,
0.8213983700428921, 2.179383177492332, 2.963589363880053, 4.039947963096991, 4.686139442532922, 5.149935606091350, 5.903646284859285, 6.559171157434498, 7.440212654826769, 7.675937734443965, 8.536017817456366, 9.186398647822894, 10.14544997134552, 10.50403052934678, 11.25602391923793, 11.67278362934388, 12.01115766788416, 12.78469027864833, 13.16773763213170, 14.23265205943321, 14.57061407906520, 15.40485093080377, 15.73058644092489, 16.36244137166378, 16.91208332511826
