| L(s) = 1 | + 2-s + 3.30·3-s + 4-s + 1.60·5-s + 3.30·6-s − 7-s + 8-s + 7.92·9-s + 1.60·10-s − 1.00·11-s + 3.30·12-s + 3.99·13-s − 14-s + 5.30·15-s + 16-s + 4.17·17-s + 7.92·18-s + 1.60·20-s − 3.30·21-s − 1.00·22-s − 3.41·23-s + 3.30·24-s − 2.42·25-s + 3.99·26-s + 16.2·27-s − 28-s − 9.28·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.718·5-s + 1.34·6-s − 0.377·7-s + 0.353·8-s + 2.64·9-s + 0.507·10-s − 0.303·11-s + 0.954·12-s + 1.10·13-s − 0.267·14-s + 1.37·15-s + 0.250·16-s + 1.01·17-s + 1.86·18-s + 0.359·20-s − 0.721·21-s − 0.214·22-s − 0.711·23-s + 0.674·24-s − 0.484·25-s + 0.783·26-s + 3.13·27-s − 0.188·28-s − 1.72·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.561169024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.561169024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 9.28T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 6.77T + 47T^{2} \) |
| 53 | \( 1 + 0.844T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 - 1.00T + 71T^{2} \) |
| 73 | \( 1 - 4.42T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 - 0.861T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.988840782203004016375910317530, −7.84044152751627568278844797112, −6.76788488215794018538707287503, −6.12330256913384526812218484524, −5.27134265900891318835413628685, −4.22219787024026446377393407892, −3.50095399373377570906038160739, −3.08373349665684541916723743671, −2.05934266055957602405599695513, −1.49311838123544884429564688706,
1.49311838123544884429564688706, 2.05934266055957602405599695513, 3.08373349665684541916723743671, 3.50095399373377570906038160739, 4.22219787024026446377393407892, 5.27134265900891318835413628685, 6.12330256913384526812218484524, 6.76788488215794018538707287503, 7.84044152751627568278844797112, 7.988840782203004016375910317530
