| L(s) = 1 | + 2-s + 1.30·3-s + 4-s − 5-s + 1.30·6-s + 0.697·7-s + 8-s − 1.30·9-s − 10-s + 11-s + 1.30·12-s + 6.30·13-s + 0.697·14-s − 1.30·15-s + 16-s − 3.69·17-s − 1.30·18-s + 4.60·19-s − 20-s + 0.908·21-s + 22-s + 2.90·23-s + 1.30·24-s + 25-s + 6.30·26-s − 5.60·27-s + 0.697·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.752·3-s + 0.5·4-s − 0.447·5-s + 0.531·6-s + 0.263·7-s + 0.353·8-s − 0.434·9-s − 0.316·10-s + 0.301·11-s + 0.376·12-s + 1.74·13-s + 0.186·14-s − 0.336·15-s + 0.250·16-s − 0.896·17-s − 0.307·18-s + 1.05·19-s − 0.223·20-s + 0.198·21-s + 0.213·22-s + 0.606·23-s + 0.265·24-s + 0.200·25-s + 1.23·26-s − 1.07·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.493099969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.493099969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 0.697T + 71T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 7.30T + 97T^{2} \) |
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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
−7.60997155684186407723572968197, −7.38728154846066681298629759869, −6.23594943561140656061018942714, −5.83887233496711155911273628733, −4.96288219036884347543934071326, −3.99862851892922903040239489180, −3.64414265770998524740391324811, −2.86701457690197315479218648258, −1.99066043988537744338462474722, −0.937137535955367956210019968526,
0.937137535955367956210019968526, 1.99066043988537744338462474722, 2.86701457690197315479218648258, 3.64414265770998524740391324811, 3.99862851892922903040239489180, 4.96288219036884347543934071326, 5.83887233496711155911273628733, 6.23594943561140656061018942714, 7.38728154846066681298629759869, 7.60997155684186407723572968197
