| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s − 13-s + 3·17-s + 21-s + 23-s − 27-s − 5·29-s − 7·31-s + 2·33-s − 2·37-s + 39-s + 7·41-s + 11·43-s − 8·47-s + 49-s − 3·51-s − 53-s + 5·59-s − 3·61-s − 63-s + 12·67-s − 69-s − 12·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.727·17-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.928·29-s − 1.25·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s + 1.09·41-s + 1.67·43-s − 1.16·47-s + 1/7·49-s − 0.420·51-s − 0.137·53-s + 0.650·59-s − 0.384·61-s − 0.125·63-s + 1.46·67-s − 0.120·69-s − 1.42·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110932859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110932859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.44420915888115246850830981287, −7.35922470787993274922963431248, −6.29631668460529036595226032524, −5.68000902806688920598142527352, −5.18659840426284330004136835639, −4.29337610096870261749522746467, −3.52496923816068091274840534294, −2.67662282356180140639929727839, −1.69866388681464351182940481369, −0.52825552997668999429872880161,
0.52825552997668999429872880161, 1.69866388681464351182940481369, 2.67662282356180140639929727839, 3.52496923816068091274840534294, 4.29337610096870261749522746467, 5.18659840426284330004136835639, 5.68000902806688920598142527352, 6.29631668460529036595226032524, 7.35922470787993274922963431248, 7.44420915888115246850830981287
