| L(s) = 1 | + 2.78·2-s − 3-s + 5.74·4-s − 5-s − 2.78·6-s + 2.63·7-s + 10.4·8-s + 9-s − 2.78·10-s − 2.15·11-s − 5.74·12-s − 13-s + 7.31·14-s + 15-s + 17.4·16-s + 2.72·17-s + 2.78·18-s + 5.12·19-s − 5.74·20-s − 2.63·21-s − 5.98·22-s − 1.65·23-s − 10.4·24-s + 25-s − 2.78·26-s − 27-s + 15.1·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.87·4-s − 0.447·5-s − 1.13·6-s + 0.994·7-s + 3.68·8-s + 0.333·9-s − 0.879·10-s − 0.648·11-s − 1.65·12-s − 0.277·13-s + 1.95·14-s + 0.258·15-s + 4.36·16-s + 0.661·17-s + 0.655·18-s + 1.17·19-s − 1.28·20-s − 0.574·21-s − 1.27·22-s − 0.345·23-s − 2.12·24-s + 0.200·25-s − 0.545·26-s − 0.192·27-s + 2.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.125107161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.125107161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 - 0.589T + 29T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 + 6.17T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 - 0.514T + 79T^{2} \) |
| 83 | \( 1 - 3.63T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 7.90T + 97T^{2} \) |
| 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
−7.69272524848293596390712286470, −7.25550415338763607784524561589, −6.36748717537837704294569405632, −5.68688404991517247772264665187, −4.91523970207586191440960061846, −4.82171653399977177630342649047, −3.77087628939888538000664509174, −3.09855351290648096112534154986, −2.13655512415326538951405966813, −1.16556344421679427988011756421,
1.16556344421679427988011756421, 2.13655512415326538951405966813, 3.09855351290648096112534154986, 3.77087628939888538000664509174, 4.82171653399977177630342649047, 4.91523970207586191440960061846, 5.68688404991517247772264665187, 6.36748717537837704294569405632, 7.25550415338763607784524561589, 7.69272524848293596390712286470
