| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 5·11-s − 13-s − 15-s + 3·17-s − 6·19-s − 21-s + 6·23-s + 25-s + 5·27-s + 9·29-s + 5·33-s + 35-s − 6·37-s + 39-s + 8·41-s + 6·43-s − 2·45-s − 3·47-s + 49-s − 3·51-s + 12·53-s − 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 1.37·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 0.870·33-s + 0.169·35-s − 0.986·37-s + 0.160·39-s + 1.24·41-s + 0.914·43-s − 0.298·45-s − 0.437·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267433643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267433643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.830045079895624675979380363550, −8.389727313066474394234784453414, −7.47320461628172485062800558626, −6.62516017863013169349321112564, −5.71012569325521424791956643699, −5.22465963352202663879592762973, −4.44994957570517170152202331319, −2.98004661899199231115437847100, −2.31436414126049041586401831975, −0.73530382675733263946372082289,
0.73530382675733263946372082289, 2.31436414126049041586401831975, 2.98004661899199231115437847100, 4.44994957570517170152202331319, 5.22465963352202663879592762973, 5.71012569325521424791956643699, 6.62516017863013169349321112564, 7.47320461628172485062800558626, 8.389727313066474394234784453414, 8.830045079895624675979380363550
