| L(s) = 1 | + 3-s + 3.85·5-s + 3.18·7-s + 9-s − 4.63·11-s + 4.72·13-s + 3.85·15-s − 2.60·17-s − 0.619·19-s + 3.18·21-s − 2.05·23-s + 9.84·25-s + 27-s + 4.63·29-s − 6.28·31-s − 4.63·33-s + 12.2·35-s + 0.540·37-s + 4.72·39-s + 6.06·41-s − 5.28·43-s + 3.85·45-s + 2.81·47-s + 3.12·49-s − 2.60·51-s + 13.5·53-s − 17.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.72·5-s + 1.20·7-s + 0.333·9-s − 1.39·11-s + 1.30·13-s + 0.994·15-s − 0.632·17-s − 0.142·19-s + 0.694·21-s − 0.429·23-s + 1.96·25-s + 0.192·27-s + 0.861·29-s − 1.12·31-s − 0.806·33-s + 2.07·35-s + 0.0889·37-s + 0.756·39-s + 0.947·41-s − 0.806·43-s + 0.574·45-s + 0.411·47-s + 0.446·49-s − 0.365·51-s + 1.86·53-s − 2.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.173075730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.173075730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 4.72T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 0.619T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 - 0.540T + 37T^{2} \) |
| 41 | \( 1 - 6.06T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 2.81T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 7.98T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.148667455668611027917282917207, −7.53831970656238570212197119100, −6.53303658206523122052233824438, −5.89461306202848123943687900361, −5.23099247397935665540087350626, −4.62992190806796590691926895631, −3.54191644852201220153063903513, −2.40799328593491056921831682012, −2.06260596271446424966441331556, −1.12530371237935284011982754108,
1.12530371237935284011982754108, 2.06260596271446424966441331556, 2.40799328593491056921831682012, 3.54191644852201220153063903513, 4.62992190806796590691926895631, 5.23099247397935665540087350626, 5.89461306202848123943687900361, 6.53303658206523122052233824438, 7.53831970656238570212197119100, 8.148667455668611027917282917207
