| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s + 19-s + 21-s − 3·23-s − 4·25-s − 27-s − 5·29-s + 9·31-s + 2·33-s + 35-s − 2·41-s + 43-s − 45-s + 3·47-s + 49-s − 9·53-s + 2·55-s − 57-s − 2·61-s − 63-s + 10·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s + 1.61·31-s + 0.348·33-s + 0.169·35-s − 0.312·41-s + 0.152·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s − 1.23·53-s + 0.269·55-s − 0.132·57-s − 0.256·61-s − 0.125·63-s + 1.22·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5752774235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5752774235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.40034432631267, −13.78857174163669, −13.19845246391241, −12.97957403431803, −12.18382641206145, −11.79254305480230, −11.52216636659775, −10.67656922839172, −10.41190531724601, −9.774621961627515, −9.361470288079016, −8.614794854576367, −7.981185113154679, −7.634493184417992, −7.029173697280145, −6.361415992278516, −5.894247094948070, −5.375222349809926, −4.636092009923224, −4.182686756512651, −3.479155125008865, −2.845682596435437, −2.081527457379423, −1.248979155476029, −0.2835377398057606,
0.2835377398057606, 1.248979155476029, 2.081527457379423, 2.845682596435437, 3.479155125008865, 4.182686756512651, 4.636092009923224, 5.375222349809926, 5.894247094948070, 6.361415992278516, 7.029173697280145, 7.634493184417992, 7.981185113154679, 8.614794854576367, 9.361470288079016, 9.774621961627515, 10.41190531724601, 10.67656922839172, 11.52216636659775, 11.79254305480230, 12.18382641206145, 12.97957403431803, 13.19845246391241, 13.78857174163669, 14.40034432631267
