| L(s) = 1 | − 2·3-s + 3·5-s − 7-s + 9-s − 6·11-s − 6·15-s − 3·17-s − 4·19-s + 2·21-s + 6·23-s + 4·25-s + 4·27-s − 3·29-s + 10·31-s + 12·33-s − 3·35-s + 37-s − 3·41-s − 10·43-s + 3·45-s + 49-s + 6·51-s + 9·53-s − 18·55-s + 8·57-s − 6·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.54·15-s − 0.727·17-s − 0.917·19-s + 0.436·21-s + 1.25·23-s + 4/5·25-s + 0.769·27-s − 0.557·29-s + 1.79·31-s + 2.08·33-s − 0.507·35-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.447·45-s + 1/7·49-s + 0.840·51-s + 1.23·53-s − 2.42·55-s + 1.05·57-s − 0.781·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6611951503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6611951503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.90667895793565, −13.25319351253005, −13.09474274955595, −12.83284513939835, −12.00148570497607, −11.50445616937326, −10.96114700873878, −10.52538375232501, −10.03684769331377, −9.910636450903631, −8.932442898067909, −8.568177838979290, −8.001406401974867, −6.958438610678187, −6.877603336976626, −6.070537113964569, −5.808433581248270, −5.206700897603964, −4.858420857784082, −4.246381584057942, −3.046835753725486, −2.688119117083492, −2.071610727409935, −1.223434243694052, −0.2932463175729376,
0.2932463175729376, 1.223434243694052, 2.071610727409935, 2.688119117083492, 3.046835753725486, 4.246381584057942, 4.858420857784082, 5.206700897603964, 5.808433581248270, 6.070537113964569, 6.877603336976626, 6.958438610678187, 8.001406401974867, 8.568177838979290, 8.932442898067909, 9.910636450903631, 10.03684769331377, 10.52538375232501, 10.96114700873878, 11.50445616937326, 12.00148570497607, 12.83284513939835, 13.09474274955595, 13.25319351253005, 13.90667895793565
