| L(s) = 1 | − 3.18·3-s + 5-s − 7-s + 7.11·9-s + 1.30·11-s + 4.75·13-s − 3.18·15-s + 6.06·17-s − 2.98·19-s + 3.18·21-s + 6.20·23-s + 25-s − 13.0·27-s + 8.60·29-s − 0.990·31-s − 4.14·33-s − 35-s + 6.56·37-s − 15.1·39-s + 8.73·41-s + 43-s + 7.11·45-s + 4.91·47-s + 49-s − 19.2·51-s + 1.76·53-s + 1.30·55-s + ⋯ |
| L(s) = 1 | − 1.83·3-s + 0.447·5-s − 0.377·7-s + 2.37·9-s + 0.392·11-s + 1.31·13-s − 0.821·15-s + 1.46·17-s − 0.684·19-s + 0.694·21-s + 1.29·23-s + 0.200·25-s − 2.51·27-s + 1.59·29-s − 0.177·31-s − 0.720·33-s − 0.169·35-s + 1.07·37-s − 2.42·39-s + 1.36·41-s + 0.152·43-s + 1.06·45-s + 0.716·47-s + 0.142·49-s − 2.69·51-s + 0.241·53-s + 0.175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.465119777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465119777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 0.990T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 8.73T + 41T^{2} \) |
| 47 | \( 1 - 4.91T + 47T^{2} \) |
| 53 | \( 1 - 1.76T + 53T^{2} \) |
| 59 | \( 1 - 8.52T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.950808492627533843562694058714, −7.00949378190690572127092097028, −6.50430173049771241821264680353, −5.87932680662034028774325532970, −5.50590582962962907197574813235, −4.56637667063701033113555722284, −3.90761382793375311717468789483, −2.80462712433257673506177365112, −1.28362872993812253685270950275, −0.850781597888523430588944917524,
0.850781597888523430588944917524, 1.28362872993812253685270950275, 2.80462712433257673506177365112, 3.90761382793375311717468789483, 4.56637667063701033113555722284, 5.50590582962962907197574813235, 5.87932680662034028774325532970, 6.50430173049771241821264680353, 7.00949378190690572127092097028, 7.950808492627533843562694058714
