| L(s) = 1 | − 2-s − 0.878·3-s + 4-s + 5-s + 0.878·6-s − 7-s − 8-s − 2.22·9-s − 10-s + 5.22·11-s − 0.878·12-s + 14-s − 0.878·15-s + 16-s − 17-s + 2.22·18-s − 1.22·19-s + 20-s + 0.878·21-s − 5.22·22-s − 0.878·23-s + 0.878·24-s + 25-s + 4.59·27-s − 28-s − 0.878·29-s + 0.878·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.507·3-s + 0.5·4-s + 0.447·5-s + 0.358·6-s − 0.377·7-s − 0.353·8-s − 0.742·9-s − 0.316·10-s + 1.57·11-s − 0.253·12-s + 0.267·14-s − 0.226·15-s + 0.250·16-s − 0.242·17-s + 0.525·18-s − 0.281·19-s + 0.223·20-s + 0.191·21-s − 1.11·22-s − 0.183·23-s + 0.179·24-s + 0.200·25-s + 0.883·27-s − 0.188·28-s − 0.163·29-s + 0.160·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9880056098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9880056098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.878T + 3T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + 0.878T + 23T^{2} \) |
| 29 | \( 1 + 0.878T + 29T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 9.22T + 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 + 0.878T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 5.51T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 5.51T + 97T^{2} \) |
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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
−9.666251439807998548495797076890, −9.019776590791979577304514568772, −8.358253586923520259878039094080, −7.17831450247848900647771123196, −6.30819548665647340463893864846, −5.96850736466083950411423849905, −4.65507753379730849471058722559, −3.44797117110380653063793151612, −2.22956407793659002841364302374, −0.855482116344376919307439639396,
0.855482116344376919307439639396, 2.22956407793659002841364302374, 3.44797117110380653063793151612, 4.65507753379730849471058722559, 5.96850736466083950411423849905, 6.30819548665647340463893864846, 7.17831450247848900647771123196, 8.358253586923520259878039094080, 9.019776590791979577304514568772, 9.666251439807998548495797076890
