| L(s) = 1 | − 3.32·3-s + 5-s + 5.02·7-s + 8.02·9-s − 1.70·11-s + 13-s − 3.32·15-s − 4.64·17-s + 4.34·19-s − 16.6·21-s + 0.679·23-s + 25-s − 16.6·27-s − 1.02·29-s + 2.29·31-s + 5.67·33-s + 5.02·35-s − 1.61·37-s − 3.32·39-s + 4.64·41-s + 3.32·43-s + 8.02·45-s − 1.02·47-s + 18.2·49-s + 15.4·51-s − 9.41·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | − 1.91·3-s + 0.447·5-s + 1.90·7-s + 2.67·9-s − 0.514·11-s + 0.277·13-s − 0.857·15-s − 1.12·17-s + 0.997·19-s − 3.64·21-s + 0.141·23-s + 0.200·25-s − 3.21·27-s − 0.190·29-s + 0.411·31-s + 0.987·33-s + 0.849·35-s − 0.265·37-s − 0.531·39-s + 0.724·41-s + 0.506·43-s + 1.19·45-s − 0.149·47-s + 2.61·49-s + 2.15·51-s − 1.29·53-s − 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.151927160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151927160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 7 | \( 1 - 5.02T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 0.679T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 + 9.41T + 53T^{2} \) |
| 59 | \( 1 - 8.93T + 59T^{2} \) |
| 61 | \( 1 - 9.02T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 8.25T + 83T^{2} \) |
| 89 | \( 1 - 1.22T + 89T^{2} \) |
| 97 | \( 1 + 0.0565T + 97T^{2} \) |
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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
−10.30326757600797954963713516982, −9.258438130983416402522223497042, −8.073128485166722362977097035440, −7.28349743492730106078905851985, −6.37295266692506738541710135767, −5.42166531150596014015407972509, −5.00722323173259323739389703846, −4.21443013578930205411355421294, −2.03423890192733980233432499430, −0.974647290195565564127847204876,
0.974647290195565564127847204876, 2.03423890192733980233432499430, 4.21443013578930205411355421294, 5.00722323173259323739389703846, 5.42166531150596014015407972509, 6.37295266692506738541710135767, 7.28349743492730106078905851985, 8.073128485166722362977097035440, 9.258438130983416402522223497042, 10.30326757600797954963713516982
