| L(s) = 1 | − 0.193·2-s + 3-s − 1.96·4-s + 5-s − 0.193·6-s + 3.35·7-s + 0.768·8-s + 9-s − 0.193·10-s + 11-s − 1.96·12-s + 2.96·13-s − 0.649·14-s + 15-s + 3.77·16-s − 4.57·17-s − 0.193·18-s − 4.31·19-s − 1.96·20-s + 3.35·21-s − 0.193·22-s − 6.70·23-s + 0.768·24-s + 25-s − 0.574·26-s + 27-s − 6.57·28-s + ⋯ |
| L(s) = 1 | − 0.137·2-s + 0.577·3-s − 0.981·4-s + 0.447·5-s − 0.0791·6-s + 1.26·7-s + 0.271·8-s + 0.333·9-s − 0.0613·10-s + 0.301·11-s − 0.566·12-s + 0.821·13-s − 0.173·14-s + 0.258·15-s + 0.943·16-s − 1.10·17-s − 0.0457·18-s − 0.989·19-s − 0.438·20-s + 0.731·21-s − 0.0413·22-s − 1.39·23-s + 0.156·24-s + 0.200·25-s − 0.112·26-s + 0.192·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237765003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237765003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 + 9.27T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 - 0.0752T + 97T^{2} \) |
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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
−13.21934108488699740783375001310, −11.83832033647563464082643787492, −10.67583053527619316221666719387, −9.677502694457118962578811900873, −8.482778303087885725523643480396, −8.225417110936899030207095146560, −6.44595563315987795679485687678, −4.93818317682541769973429076548, −3.96023019054470350865869032629, −1.82173589598164427283868781627,
1.82173589598164427283868781627, 3.96023019054470350865869032629, 4.93818317682541769973429076548, 6.44595563315987795679485687678, 8.225417110936899030207095146560, 8.482778303087885725523643480396, 9.677502694457118962578811900873, 10.67583053527619316221666719387, 11.83832033647563464082643787492, 13.21934108488699740783375001310
