| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 0.0952·13-s − 15-s − 3.30·17-s + 5.30·19-s − 21-s + 3.30·23-s + 25-s + 27-s + 5.39·29-s + 0.0952·31-s + 33-s + 35-s − 0.0952·37-s − 0.0952·39-s + 2.09·41-s − 9.99·43-s − 45-s − 8.69·47-s + 49-s − 3.30·51-s − 7.30·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.0264·13-s − 0.258·15-s − 0.800·17-s + 1.21·19-s − 0.218·21-s + 0.688·23-s + 0.200·25-s + 0.192·27-s + 1.00·29-s + 0.0171·31-s + 0.174·33-s + 0.169·35-s − 0.0156·37-s − 0.0152·39-s + 0.327·41-s − 1.52·43-s − 0.149·45-s − 1.26·47-s + 0.142·49-s − 0.462·51-s − 1.00·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.120997995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120997995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 0.0952T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 0.0952T + 31T^{2} \) |
| 37 | \( 1 + 0.0952T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 + 9.99T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 - 5.30T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.09T + 79T^{2} \) |
| 83 | \( 1 - 1.30T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−8.313728024060583228736802633613, −7.67488930817479423536239057162, −6.83608227165400751992234012024, −6.41601230006125628446069652010, −5.18396561105718170526601997910, −4.59693293609385717873831264541, −3.55541049599526426508412334892, −3.08646461871416864961231704385, −2.00492115737313492578562928884, −0.792432245129186541155956577782,
0.792432245129186541155956577782, 2.00492115737313492578562928884, 3.08646461871416864961231704385, 3.55541049599526426508412334892, 4.59693293609385717873831264541, 5.18396561105718170526601997910, 6.41601230006125628446069652010, 6.83608227165400751992234012024, 7.67488930817479423536239057162, 8.313728024060583228736802633613
