| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 3·11-s − 13-s − 4·14-s + 16-s + 19-s − 20-s − 3·22-s + 7·23-s + 25-s + 26-s + 4·28-s − 29-s + 3·31-s − 32-s − 4·35-s + 11·37-s − 38-s + 40-s − 4·41-s + 7·43-s + 3·44-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 + 0.904·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s − 0.639·22-s + 1.45·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s − 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.676·35-s + 1.80·37-s − 0.162·38-s + 0.158·40-s − 0.624·41-s + 1.06·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054414958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054414958\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.36485944049876, −11.91242602483551, −11.67710888286860, −11.11561201649092, −10.84089244536549, −10.51917092081811, −9.624969337081318, −9.345427058229339, −8.998650498256131, −8.324205899532203, −8.063547684862470, −7.577767629449786, −7.227385858707758, −6.579083418639629, −6.252759105469462, −5.408446896821802, −5.078088792462315, −4.509899490066355, −4.051451368773467, −3.475338068615687, −2.680087924711892, −2.334017046255439, −1.481129093117967, −1.114958289863420, −0.5874014078943369,
0.5874014078943369, 1.114958289863420, 1.481129093117967, 2.334017046255439, 2.680087924711892, 3.475338068615687, 4.051451368773467, 4.509899490066355, 5.078088792462315, 5.408446896821802, 6.252759105469462, 6.579083418639629, 7.227385858707758, 7.577767629449786, 8.063547684862470, 8.324205899532203, 8.998650498256131, 9.345427058229339, 9.624969337081318, 10.51917092081811, 10.84089244536549, 11.11561201649092, 11.67710888286860, 11.91242602483551, 12.36485944049876
