| L(s) = 1 | − 2·3-s − 2·5-s + 9-s − 2·13-s + 4·15-s + 2·17-s − 25-s + 4·27-s − 8·29-s − 8·31-s + 4·37-s + 4·39-s + 10·41-s − 2·43-s − 2·45-s − 7·49-s − 4·51-s − 8·53-s − 14·59-s + 4·65-s − 8·67-s − 8·71-s − 6·73-s + 2·75-s − 79-s − 11·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.485·17-s − 1/5·25-s + 0.769·27-s − 1.48·29-s − 1.43·31-s + 0.657·37-s + 0.640·39-s + 1.56·41-s − 0.304·43-s − 0.298·45-s − 49-s − 0.560·51-s − 1.09·53-s − 1.82·59-s + 0.496·65-s − 0.977·67-s − 0.949·71-s − 0.702·73-s + 0.230·75-s − 0.112·79-s − 1.22·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.33688719654188, −12.93902179078633, −12.50160307544349, −12.03971448589307, −11.62133667586735, −11.29141315598576, −10.73101526278123, −10.54565335491738, −9.648852159152618, −9.336616848549536, −8.836079070363281, −7.981019194590400, −7.568506418196528, −7.446401702281809, −6.597008605674519, −6.093403653801213, −5.674521464565246, −5.177627226183858, −4.540996240752570, −4.199515613647626, −3.395706036135395, −3.037164078788368, −2.064956985549067, −1.447493330612932, −0.5182740885621592, 0,
0.5182740885621592, 1.447493330612932, 2.064956985549067, 3.037164078788368, 3.395706036135395, 4.199515613647626, 4.540996240752570, 5.177627226183858, 5.674521464565246, 6.093403653801213, 6.597008605674519, 7.446401702281809, 7.568506418196528, 7.981019194590400, 8.836079070363281, 9.336616848549536, 9.648852159152618, 10.54565335491738, 10.73101526278123, 11.29141315598576, 11.62133667586735, 12.03971448589307, 12.50160307544349, 12.93902179078633, 13.33688719654188
