| L(s) = 1 | − 3·3-s + 20·7-s + 9·9-s + 56·11-s + 86·13-s + 106·17-s − 4·19-s − 60·21-s + 136·23-s − 27·27-s − 206·29-s + 152·31-s − 168·33-s − 282·37-s − 258·39-s − 246·41-s + 412·43-s + 40·47-s + 57·49-s − 318·51-s + 126·53-s + 12·57-s − 56·59-s − 2·61-s + 180·63-s − 388·67-s − 408·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.07·7-s + 1/3·9-s + 1.53·11-s + 1.83·13-s + 1.51·17-s − 0.0482·19-s − 0.623·21-s + 1.23·23-s − 0.192·27-s − 1.31·29-s + 0.880·31-s − 0.886·33-s − 1.25·37-s − 1.05·39-s − 0.937·41-s + 1.46·43-s + 0.124·47-s + 0.166·49-s − 0.873·51-s + 0.326·53-s + 0.0278·57-s − 0.123·59-s − 0.00419·61-s + 0.359·63-s − 0.707·67-s − 0.711·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.831537889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831537889\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 136 T + p^{3} T^{2} \) |
| 29 | \( 1 + 206 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 282 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 40 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 + 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 + 388 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 + 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 668 T + p^{3} T^{2} \) |
| 89 | \( 1 - 66 T + p^{3} T^{2} \) |
| 97 | \( 1 - 926 T + p^{3} T^{2} \) |
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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
−9.228719474123424507726242394655, −8.642787086567515447832854322467, −7.72512522668228821112407820153, −6.80555026213817265309981717761, −5.96209368935814840022399541953, −5.23229755374600752314305511168, −4.13833249537026104316564842731, −3.40081314787443256175409763184, −1.51591072113104370742593254817, −1.07425156619760232027538579515,
1.07425156619760232027538579515, 1.51591072113104370742593254817, 3.40081314787443256175409763184, 4.13833249537026104316564842731, 5.23229755374600752314305511168, 5.96209368935814840022399541953, 6.80555026213817265309981717761, 7.72512522668228821112407820153, 8.642787086567515447832854322467, 9.228719474123424507726242394655
