| L(s) = 1 | − 2.73·2-s − 3-s + 5.46·4-s − 5-s + 2.73·6-s + 3·7-s − 9.46·8-s + 9-s + 2.73·10-s + 2.73·11-s − 5.46·12-s + 6.19·13-s − 8.19·14-s + 15-s + 14.9·16-s + 2.26·17-s − 2.73·18-s + 3.26·19-s − 5.46·20-s − 3·21-s − 7.46·22-s + 9.46·24-s + 25-s − 16.9·26-s − 27-s + 16.3·28-s − 3.73·29-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.73·4-s − 0.447·5-s + 1.11·6-s + 1.13·7-s − 3.34·8-s + 0.333·9-s + 0.863·10-s + 0.823·11-s − 1.57·12-s + 1.71·13-s − 2.19·14-s + 0.258·15-s + 3.73·16-s + 0.550·17-s − 0.643·18-s + 0.749·19-s − 1.22·20-s − 0.654·21-s − 1.59·22-s + 1.93·24-s + 0.200·25-s − 3.31·26-s − 0.192·27-s + 3.09·28-s − 0.693·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9556675672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9556675672\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.46T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.960518349630987529297759616541, −7.29997315482212741663806021559, −6.86679949237979957944881805346, −5.82461747067285448968368699590, −5.54363928452829240993556808593, −4.07295125944214736102858855321, −3.43594701889800683936114306601, −2.10228018708179518995071166976, −1.31179861738137427577568798831, −0.791162678959879539951172091484,
0.791162678959879539951172091484, 1.31179861738137427577568798831, 2.10228018708179518995071166976, 3.43594701889800683936114306601, 4.07295125944214736102858855321, 5.54363928452829240993556808593, 5.82461747067285448968368699590, 6.86679949237979957944881805346, 7.29997315482212741663806021559, 7.960518349630987529297759616541
