| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 6·13-s + 15-s + 2·17-s − 4·19-s + 21-s − 23-s + 25-s − 27-s + 2·29-s − 4·31-s + 35-s − 2·37-s − 6·39-s − 6·41-s + 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s − 10·53-s + 4·57-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.169·35-s − 0.328·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490583824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490583824\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.19043868137353, −12.84729972327070, −12.29781876999360, −11.95109216747875, −11.33604558847129, −10.94089761101002, −10.52753633204867, −10.18430358790396, −9.370488243767660, −9.016797787276451, −8.452387972986209, −8.000259244108380, −7.488694779415707, −6.786627331623360, −6.425276340900713, −5.981522427423740, −5.455289336152607, −4.843167251567959, −4.219277759325386, −3.644990687482411, −3.411381693396768, −2.493158737664930, −1.763937467881671, −1.093241113572506, −0.4202849553883782,
0.4202849553883782, 1.093241113572506, 1.763937467881671, 2.493158737664930, 3.411381693396768, 3.644990687482411, 4.219277759325386, 4.843167251567959, 5.455289336152607, 5.981522427423740, 6.425276340900713, 6.786627331623360, 7.488694779415707, 8.000259244108380, 8.452387972986209, 9.016797787276451, 9.370488243767660, 10.18430358790396, 10.52753633204867, 10.94089761101002, 11.33604558847129, 11.95109216747875, 12.29781876999360, 12.84729972327070, 13.19043868137353
