| L(s) = 1 | − 42.7·2-s + 1.31e3·4-s − 625·5-s − 2.40e3·7-s − 3.44e4·8-s + 2.67e4·10-s − 4.15e4·11-s + 1.03e5·13-s + 1.02e5·14-s + 8.00e5·16-s − 3.55e5·17-s − 3.49e5·19-s − 8.23e5·20-s + 1.77e6·22-s − 2.28e5·23-s + 3.90e5·25-s − 4.44e6·26-s − 3.16e6·28-s − 4.02e6·29-s − 3.29e6·31-s − 1.65e7·32-s + 1.51e7·34-s + 1.50e6·35-s − 2.13e7·37-s + 1.49e7·38-s + 2.15e7·40-s − 1.05e7·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.57·4-s − 0.447·5-s − 0.377·7-s − 2.97·8-s + 0.845·10-s − 0.856·11-s + 1.00·13-s + 0.714·14-s + 3.05·16-s − 1.03·17-s − 0.614·19-s − 1.15·20-s + 1.61·22-s − 0.170·23-s + 0.200·25-s − 1.90·26-s − 0.973·28-s − 1.05·29-s − 0.640·31-s − 2.79·32-s + 1.95·34-s + 0.169·35-s − 1.86·37-s + 1.16·38-s + 1.33·40-s − 0.581·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1879752958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1879752958\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 42.7T + 512T^{2} \) |
| 11 | \( 1 + 4.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.03e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.49e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.28e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.13e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.05e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.89e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.31e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.31e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.86e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.37e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.36e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.74e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.75e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.76e8T + 7.60e17T^{2} \) |
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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
−10.10010272566521228734004060608, −8.892521828881546059602082796421, −8.537535677376037458021480655571, −7.45041351000774092201192469335, −6.74204442724930753785075941579, −5.65169859744244241383429003622, −3.77693023138087044460310282331, −2.54034599117574784892349132871, −1.55603833284047938926997592932, −0.25020778936009660289781570100,
0.25020778936009660289781570100, 1.55603833284047938926997592932, 2.54034599117574784892349132871, 3.77693023138087044460310282331, 5.65169859744244241383429003622, 6.74204442724930753785075941579, 7.45041351000774092201192469335, 8.537535677376037458021480655571, 8.892521828881546059602082796421, 10.10010272566521228734004060608
