| L(s) = 1 | − 4.57·3-s + 5·5-s − 22.7·7-s − 6.03·9-s − 45.6·11-s + 77.0·13-s − 22.8·15-s − 5.11·17-s − 53.0·19-s + 104.·21-s + 23·23-s + 25·25-s + 151.·27-s − 233.·29-s − 151.·31-s + 209.·33-s − 113.·35-s − 115.·37-s − 353.·39-s − 488.·41-s − 405.·43-s − 30.1·45-s + 43.9·47-s + 174.·49-s + 23.4·51-s + 130.·53-s − 228.·55-s + ⋯ |
| L(s) = 1 | − 0.881·3-s + 0.447·5-s − 1.22·7-s − 0.223·9-s − 1.25·11-s + 1.64·13-s − 0.394·15-s − 0.0730·17-s − 0.639·19-s + 1.08·21-s + 0.208·23-s + 0.200·25-s + 1.07·27-s − 1.49·29-s − 0.879·31-s + 1.10·33-s − 0.549·35-s − 0.512·37-s − 1.44·39-s − 1.86·41-s − 1.43·43-s − 0.0999·45-s + 0.136·47-s + 0.507·49-s + 0.0643·51-s + 0.339·53-s − 0.560·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5254091006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5254091006\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 4.57T + 27T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 11 | \( 1 + 45.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.11T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 488.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 43.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 130.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 821.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 505.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 561.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 885.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 884.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 160.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 819.T + 9.12e5T^{2} \) |
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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
−8.844806279534787060137714240517, −8.275623667490358245369151169391, −6.98646361328951122526562359951, −6.41766409719681359316502558756, −5.65686541325601710297063425846, −5.20390314606398717770063285915, −3.77525912120852390635536049853, −3.03670636158593634557039351376, −1.78743431817989618476250362556, −0.34205676330118108578344058779,
0.34205676330118108578344058779, 1.78743431817989618476250362556, 3.03670636158593634557039351376, 3.77525912120852390635536049853, 5.20390314606398717770063285915, 5.65686541325601710297063425846, 6.41766409719681359316502558756, 6.98646361328951122526562359951, 8.275623667490358245369151169391, 8.844806279534787060137714240517
