| L(s) = 1 | + 2-s + 2.39·3-s + 4-s − 0.391·5-s + 2.39·6-s − 1.71·7-s + 8-s + 2.71·9-s − 0.391·10-s − 11-s + 2.39·12-s + 3.71·13-s − 1.71·14-s − 0.935·15-s + 16-s + 5.43·17-s + 2.71·18-s − 19-s − 0.391·20-s − 4.11·21-s − 22-s − 3.43·23-s + 2.39·24-s − 4.84·25-s + 3.71·26-s − 0.672·27-s − 1.71·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.38·3-s + 0.5·4-s − 0.175·5-s + 0.976·6-s − 0.649·7-s + 0.353·8-s + 0.906·9-s − 0.123·10-s − 0.301·11-s + 0.690·12-s + 1.03·13-s − 0.459·14-s − 0.241·15-s + 0.250·16-s + 1.31·17-s + 0.640·18-s − 0.229·19-s − 0.0875·20-s − 0.896·21-s − 0.213·22-s − 0.716·23-s + 0.488·24-s − 0.969·25-s + 0.729·26-s − 0.129·27-s − 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.896988465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.896988465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 0.391T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 - 1.04T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 8.51T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
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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
−11.31468820881441681215976561818, −10.16050325440825332556907548324, −9.400503517462249451900216195901, −8.275157492226730483224248510818, −7.69558298026064181626568328514, −6.46899371741935064110887626858, −5.41305306052668382208684587190, −3.75176066846497908511832953172, −3.38782446174387078263704721746, −1.99354725019068991302511362310,
1.99354725019068991302511362310, 3.38782446174387078263704721746, 3.75176066846497908511832953172, 5.41305306052668382208684587190, 6.46899371741935064110887626858, 7.69558298026064181626568328514, 8.275157492226730483224248510818, 9.400503517462249451900216195901, 10.16050325440825332556907548324, 11.31468820881441681215976561818
