L(s) = 1 | − 3-s − 2·4-s − 3.46·5-s − 1.73·7-s + 9-s + 3.46·11-s + 2·12-s + 3.46·15-s + 4·16-s − 3.46·19-s + 6.92·20-s + 1.73·21-s + 6·23-s + 6.99·25-s − 27-s + 3.46·28-s + 6·29-s − 1.73·31-s − 3.46·33-s + 5.99·35-s − 2·36-s − 6.92·41-s + 43-s − 6.92·44-s − 3.46·45-s + 3.46·47-s − 4·48-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 1.54·5-s − 0.654·7-s + 0.333·9-s + 1.04·11-s + 0.577·12-s + 0.894·15-s + 16-s − 0.794·19-s + 1.54·20-s + 0.377·21-s + 1.25·23-s + 1.39·25-s − 0.192·27-s + 0.654·28-s + 1.11·29-s − 0.311·31-s − 0.603·33-s + 1.01·35-s − 0.333·36-s − 1.08·41-s + 0.152·43-s − 1.04·44-s − 0.516·45-s + 0.505·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5579216933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5579216933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−11.00843776604560594061261593158, −10.05002789396545129821501583065, −9.001977522851385071618946957361, −8.373019181015416354772935714092, −7.22492522500122658860855616044, −6.40391621913323918357165726223, −5.01325190889063099099130453845, −4.15327892049719793374487707532, −3.40682422403086942274257999549, −0.70128096592488216986894483706,
0.70128096592488216986894483706, 3.40682422403086942274257999549, 4.15327892049719793374487707532, 5.01325190889063099099130453845, 6.40391621913323918357165726223, 7.22492522500122658860855616044, 8.373019181015416354772935714092, 9.001977522851385071618946957361, 10.05002789396545129821501583065, 11.00843776604560594061261593158