| L(s) = 1 | − 3-s − 1.30·5-s − 1.53·7-s + 9-s − 1.10·11-s + 4.17·13-s + 1.30·15-s − 4.29·17-s − 5.98·19-s + 1.53·21-s + 3.64·23-s − 3.30·25-s − 27-s + 4.89·29-s + 2.10·31-s + 1.10·33-s + 2.00·35-s + 5.69·37-s − 4.17·39-s + 2.38·41-s − 4.81·43-s − 1.30·45-s − 5.53·47-s − 4.62·49-s + 4.29·51-s − 8.47·53-s + 1.44·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.582·5-s − 0.581·7-s + 0.333·9-s − 0.333·11-s + 1.15·13-s + 0.336·15-s − 1.04·17-s − 1.37·19-s + 0.335·21-s + 0.759·23-s − 0.660·25-s − 0.192·27-s + 0.908·29-s + 0.377·31-s + 0.192·33-s + 0.339·35-s + 0.935·37-s − 0.667·39-s + 0.373·41-s − 0.733·43-s − 0.194·45-s − 0.807·47-s − 0.661·49-s + 0.601·51-s − 1.16·53-s + 0.194·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8125648463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8125648463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 5.98T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 2.10T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 + 5.53T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 + 0.549T + 61T^{2} \) |
| 67 | \( 1 + 5.13T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 4.52T + 97T^{2} \) |
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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
−7.944836691729558522929054868215, −6.94451249746774789206348240455, −6.36523589376258609666927733106, −6.03840891239532478741334706556, −4.87190498381986090471255497494, −4.37497295267021974592707066698, −3.59538608658470482949112252068, −2.75956666300957620598338030066, −1.66856625613296726725585883110, −0.45672265397589110156383873435,
0.45672265397589110156383873435, 1.66856625613296726725585883110, 2.75956666300957620598338030066, 3.59538608658470482949112252068, 4.37497295267021974592707066698, 4.87190498381986090471255497494, 6.03840891239532478741334706556, 6.36523589376258609666927733106, 6.94451249746774789206348240455, 7.944836691729558522929054868215
