| L(s) = 1 | + 2-s + 4-s − 5-s + 4.50·7-s + 8-s − 10-s + 11-s + 5.98·13-s + 4.50·14-s + 16-s + 7.21·17-s + 0.192·19-s − 20-s + 22-s − 5.75·23-s + 25-s + 5.98·26-s + 4.50·28-s + 7.58·29-s − 3.94·31-s + 32-s + 7.21·34-s − 4.50·35-s − 0.955·37-s + 0.192·38-s − 40-s − 1.76·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.70·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 1.66·13-s + 1.20·14-s + 0.250·16-s + 1.75·17-s + 0.0442·19-s − 0.223·20-s + 0.213·22-s − 1.19·23-s + 0.200·25-s + 1.17·26-s + 0.850·28-s + 1.40·29-s − 0.708·31-s + 0.176·32-s + 1.23·34-s − 0.760·35-s − 0.157·37-s + 0.0313·38-s − 0.158·40-s − 0.275·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.878703617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.878703617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.50T + 7T^{2} \) |
| 13 | \( 1 - 5.98T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - 0.192T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 + 0.955T + 37T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 + 9.80T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.80895931307148170715046996905, −7.13924560405986800517849350393, −6.11208026319532898378090227339, −5.68096320736374279425321461459, −4.93299059804238003756672029414, −4.15234236078917859608101924216, −3.72083152135303852539268756779, −2.78108519165974971362356708691, −1.58601457419981511670750033247, −1.12779152168315124966084088594,
1.12779152168315124966084088594, 1.58601457419981511670750033247, 2.78108519165974971362356708691, 3.72083152135303852539268756779, 4.15234236078917859608101924216, 4.93299059804238003756672029414, 5.68096320736374279425321461459, 6.11208026319532898378090227339, 7.13924560405986800517849350393, 7.80895931307148170715046996905
