| L(s) = 1 | − 2.63·2-s + 4.91·4-s + 3.66·5-s + 1.33·7-s − 7.67·8-s − 9.64·10-s − 1.38·11-s − 4.52·13-s − 3.51·14-s + 10.3·16-s + 0.752·17-s + 1.24·19-s + 18.0·20-s + 3.63·22-s + 7.69·23-s + 8.45·25-s + 11.9·26-s + 6.58·28-s + 3.06·29-s − 6.08·31-s − 11.8·32-s − 1.97·34-s + 4.90·35-s + 11.0·37-s − 3.28·38-s − 28.1·40-s − 9.45·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.45·4-s + 1.64·5-s + 0.505·7-s − 2.71·8-s − 3.05·10-s − 0.416·11-s − 1.25·13-s − 0.940·14-s + 2.58·16-s + 0.182·17-s + 0.286·19-s + 4.03·20-s + 0.774·22-s + 1.60·23-s + 1.69·25-s + 2.33·26-s + 1.24·28-s + 0.569·29-s − 1.09·31-s − 2.10·32-s − 0.339·34-s + 0.829·35-s + 1.81·37-s − 0.532·38-s − 4.45·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9790657178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9790657178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 0.752T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 + 6.08T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−9.654622039679896200010939862269, −9.211434299280201908957218828604, −8.387339099074876320166162770505, −7.40609412711892895892106277869, −6.85587740595302669799295161026, −5.78364535383876551868843067638, −5.02973142123717095646491965370, −2.77445153092561366950296844503, −2.13612774898857849281736238678, −1.02183255747490655626022444514,
1.02183255747490655626022444514, 2.13612774898857849281736238678, 2.77445153092561366950296844503, 5.02973142123717095646491965370, 5.78364535383876551868843067638, 6.85587740595302669799295161026, 7.40609412711892895892106277869, 8.387339099074876320166162770505, 9.211434299280201908957218828604, 9.654622039679896200010939862269
