| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 5·11-s + 4·14-s + 16-s − 17-s + 7·19-s + 20-s + 5·22-s + 6·23-s + 25-s + 4·28-s − 4·29-s + 9·31-s + 32-s − 34-s + 4·35-s − 5·37-s + 7·38-s + 40-s − 2·41-s + 43-s + 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s + 0.223·20-s + 1.06·22-s + 1.25·23-s + 1/5·25-s + 0.755·28-s − 0.742·29-s + 1.61·31-s + 0.176·32-s − 0.171·34-s + 0.676·35-s − 0.821·37-s + 1.13·38-s + 0.158·40-s − 0.312·41-s + 0.152·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.47865052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.47865052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.99380792787469, −12.18915072538903, −11.83055702934482, −11.52793307395522, −11.20647816888736, −10.71303046529907, −9.966868298669588, −9.690753935950200, −9.048898367688960, −8.623250561222141, −8.148001995269694, −7.563845831581764, −7.050454410623234, −6.671541068895083, −6.161273136061208, −5.455006810582355, −5.117274943536338, −4.715112968279725, −4.223922703213839, −3.441330802424141, −3.224530355282783, −2.273192889168491, −1.840251198298970, −1.171927047483315, −0.8898468619861921,
0.8898468619861921, 1.171927047483315, 1.840251198298970, 2.273192889168491, 3.224530355282783, 3.441330802424141, 4.223922703213839, 4.715112968279725, 5.117274943536338, 5.455006810582355, 6.161273136061208, 6.671541068895083, 7.050454410623234, 7.563845831581764, 8.148001995269694, 8.623250561222141, 9.048898367688960, 9.690753935950200, 9.966868298669588, 10.71303046529907, 11.20647816888736, 11.52793307395522, 11.83055702934482, 12.18915072538903, 12.99380792787469
