| L(s) = 1 | − 1.60·2-s + 3-s + 0.577·4-s − 3.03·5-s − 1.60·6-s + 3.88·7-s + 2.28·8-s + 9-s + 4.86·10-s + 2.75·11-s + 0.577·12-s + 13-s − 6.24·14-s − 3.03·15-s − 4.82·16-s + 2.87·17-s − 1.60·18-s + 5.44·19-s − 1.74·20-s + 3.88·21-s − 4.42·22-s + 5.09·23-s + 2.28·24-s + 4.18·25-s − 1.60·26-s + 27-s + 2.24·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.577·3-s + 0.288·4-s − 1.35·5-s − 0.655·6-s + 1.46·7-s + 0.807·8-s + 0.333·9-s + 1.53·10-s + 0.830·11-s + 0.166·12-s + 0.277·13-s − 1.66·14-s − 0.782·15-s − 1.20·16-s + 0.697·17-s − 0.378·18-s + 1.24·19-s − 0.391·20-s + 0.848·21-s − 0.942·22-s + 1.06·23-s + 0.466·24-s + 0.836·25-s − 0.314·26-s + 0.192·27-s + 0.423·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427829012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427829012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 + 3.03T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 4.77T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.656T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
| 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
−8.296179713350309037337701910793, −8.099154590100110394109808196898, −7.31259782113872764629427301998, −6.78768681434432786772394065344, −5.19635818034906487190562389041, −4.58672229445593324178838610813, −3.85915008428117076401457087855, −2.91977080892394415944482384570, −1.45322262443108645423214038342, −0.934983370552949824273820217853,
0.934983370552949824273820217853, 1.45322262443108645423214038342, 2.91977080892394415944482384570, 3.85915008428117076401457087855, 4.58672229445593324178838610813, 5.19635818034906487190562389041, 6.78768681434432786772394065344, 7.31259782113872764629427301998, 8.099154590100110394109808196898, 8.296179713350309037337701910793
