| L(s) = 1 | + 3-s − 4·7-s + 9-s + 13-s − 2·17-s − 4·21-s + 27-s + 2·29-s − 4·31-s + 6·37-s + 39-s − 6·41-s + 4·43-s + 4·47-s + 9·49-s − 2·51-s − 10·53-s + 2·61-s − 4·63-s + 8·67-s + 4·71-s + 6·73-s − 8·79-s + 81-s + 8·83-s + 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 0.872·21-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.256·61-s − 0.503·63-s + 0.977·67-s + 0.474·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 0.878·83-s + 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.50677418365132, −13.90016494835906, −13.53862723274352, −12.94903080411049, −12.69685197463896, −12.17835351605583, −11.45239238982708, −10.91123487878885, −10.38050399736544, −9.777822330691403, −9.427844954533246, −8.981171318422698, −8.388427514919905, −7.808969528549487, −7.209821719535383, −6.575972516048310, −6.341012207454736, −5.598457020449118, −4.936446640006968, −4.106036360948897, −3.724126732948535, −3.045768258441214, −2.588202717101466, −1.827711505121619, −0.8761773240202957, 0,
0.8761773240202957, 1.827711505121619, 2.588202717101466, 3.045768258441214, 3.724126732948535, 4.106036360948897, 4.936446640006968, 5.598457020449118, 6.341012207454736, 6.575972516048310, 7.209821719535383, 7.808969528549487, 8.388427514919905, 8.981171318422698, 9.427844954533246, 9.777822330691403, 10.38050399736544, 10.91123487878885, 11.45239238982708, 12.17835351605583, 12.69685197463896, 12.94903080411049, 13.53862723274352, 13.90016494835906, 14.50677418365132
