| L(s) = 1 | − 2.52·2-s + 3.17·3-s + 4.39·4-s + 5-s − 8.03·6-s − 7-s − 6.06·8-s + 7.08·9-s − 2.52·10-s + 13.9·12-s + 5.63·13-s + 2.52·14-s + 3.17·15-s + 6.55·16-s + 3.53·17-s − 17.9·18-s + 2.61·19-s + 4.39·20-s − 3.17·21-s − 8.37·23-s − 19.2·24-s + 25-s − 14.2·26-s + 12.9·27-s − 4.39·28-s + 1.44·29-s − 8.03·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 1.83·3-s + 2.19·4-s + 0.447·5-s − 3.28·6-s − 0.377·7-s − 2.14·8-s + 2.36·9-s − 0.799·10-s + 4.03·12-s + 1.56·13-s + 0.676·14-s + 0.820·15-s + 1.63·16-s + 0.856·17-s − 4.22·18-s + 0.599·19-s + 0.983·20-s − 0.693·21-s − 1.74·23-s − 3.93·24-s + 0.200·25-s − 2.79·26-s + 2.49·27-s − 0.831·28-s + 0.267·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.116423279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116423279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 8.37T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 + 0.865T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 - 0.122T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 + 0.392T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−8.349871742944177216396829471825, −8.029207901857564830202656976943, −7.46402489325901678457644181340, −6.49494612418138768216315489133, −5.95853491569218757864758101061, −4.25514135027654094779501124093, −3.31449020461464632570962855702, −2.70619953136771594756337320446, −1.75216353954545075011299178602, −1.08250571204389989230873254808,
1.08250571204389989230873254808, 1.75216353954545075011299178602, 2.70619953136771594756337320446, 3.31449020461464632570962855702, 4.25514135027654094779501124093, 5.95853491569218757864758101061, 6.49494612418138768216315489133, 7.46402489325901678457644181340, 8.029207901857564830202656976943, 8.349871742944177216396829471825
