| L(s) = 1 | + 1.71·3-s + 2.50·5-s + 7-s − 0.0711·9-s + 1.68·11-s − 0.720·13-s + 4.28·15-s − 17-s + 5.17·19-s + 1.71·21-s − 2.40·23-s + 1.27·25-s − 5.25·27-s − 0.318·29-s − 4.96·31-s + 2.87·33-s + 2.50·35-s + 10.2·37-s − 1.23·39-s + 9.60·41-s + 1.30·43-s − 0.178·45-s + 2.49·47-s + 49-s − 1.71·51-s + 1.99·53-s + 4.21·55-s + ⋯ |
| L(s) = 1 | + 0.988·3-s + 1.12·5-s + 0.377·7-s − 0.0237·9-s + 0.507·11-s − 0.199·13-s + 1.10·15-s − 0.242·17-s + 1.18·19-s + 0.373·21-s − 0.500·23-s + 0.255·25-s − 1.01·27-s − 0.0591·29-s − 0.892·31-s + 0.500·33-s + 0.423·35-s + 1.68·37-s − 0.197·39-s + 1.50·41-s + 0.199·43-s − 0.0265·45-s + 0.364·47-s + 0.142·49-s − 0.239·51-s + 0.274·53-s + 0.568·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.062747681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.062747681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 0.720T + 13T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 0.318T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 - 1.30T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 - 1.99T + 53T^{2} \) |
| 59 | \( 1 - 2.29T + 59T^{2} \) |
| 61 | \( 1 - 0.620T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 - 1.93T + 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 1.87T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75404125299716640245652607738, −7.48745469248086034082586812571, −6.37419557342754850788094572655, −5.81899002089388383742185072059, −5.16490126112554723094015294371, −4.18324603150436402252668920628, −3.44669671134359768994428614126, −2.48913670782519548440134607705, −2.05679713838629584052461686289, −0.994359131850327605171344580328,
0.994359131850327605171344580328, 2.05679713838629584052461686289, 2.48913670782519548440134607705, 3.44669671134359768994428614126, 4.18324603150436402252668920628, 5.16490126112554723094015294371, 5.81899002089388383742185072059, 6.37419557342754850788094572655, 7.48745469248086034082586812571, 7.75404125299716640245652607738
