| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 3·11-s − 12-s − 13-s − 2·14-s + 15-s + 16-s − 3·17-s − 18-s + 2·19-s − 20-s − 2·21-s + 3·22-s + 3·23-s + 24-s + 25-s + 26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s + 0.639·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5757530757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5757530757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.29261013574433, −11.88899907296421, −11.54008565642618, −10.97492787893573, −10.82183212230587, −10.37978474050971, −9.780486525312582, −9.242744935868464, −8.949838437414596, −8.323833298862384, −7.826980799996606, −7.435346496026493, −7.293213940432828, −6.429611912677662, −6.134911590381217, −5.415686533458546, −5.043663183427725, −4.625171649440964, −4.002831066814796, −3.415539196601609, −2.652540571123276, −2.307457985578746, −1.552092628062435, −1.004500566517614, −0.2592262586174679,
0.2592262586174679, 1.004500566517614, 1.552092628062435, 2.307457985578746, 2.652540571123276, 3.415539196601609, 4.002831066814796, 4.625171649440964, 5.043663183427725, 5.415686533458546, 6.134911590381217, 6.429611912677662, 7.293213940432828, 7.435346496026493, 7.826980799996606, 8.323833298862384, 8.949838437414596, 9.242744935868464, 9.780486525312582, 10.37978474050971, 10.82183212230587, 10.97492787893573, 11.54008565642618, 11.88899907296421, 12.29261013574433
