| L(s) = 1 | + 0.269·2-s − 1.39·3-s − 1.92·4-s − 3.37·5-s − 0.377·6-s − 1.05·8-s − 1.04·9-s − 0.911·10-s − 1.57·11-s + 2.69·12-s − 2.85·13-s + 4.72·15-s + 3.56·16-s − 6.37·17-s − 0.281·18-s − 19-s + 6.51·20-s − 0.426·22-s + 2.28·23-s + 1.48·24-s + 6.42·25-s − 0.768·26-s + 5.65·27-s + 8.17·29-s + 1.27·30-s − 4.99·31-s + 3.08·32-s + ⋯ |
| L(s) = 1 | + 0.190·2-s − 0.807·3-s − 0.963·4-s − 1.51·5-s − 0.153·6-s − 0.374·8-s − 0.347·9-s − 0.288·10-s − 0.476·11-s + 0.778·12-s − 0.790·13-s + 1.22·15-s + 0.892·16-s − 1.54·17-s − 0.0663·18-s − 0.229·19-s + 1.45·20-s − 0.0908·22-s + 0.476·23-s + 0.302·24-s + 1.28·25-s − 0.150·26-s + 1.08·27-s + 1.51·29-s + 0.232·30-s − 0.897·31-s + 0.544·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2670860487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2670860487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.269T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.539T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 - 0.976T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 - 0.692T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−10.33567284921376614507540719534, −8.930855992037741878883741687395, −8.552279691581675412021506433262, −7.52014255219375669390744872053, −6.68362021617558605519586398154, −5.44520935012477249235218645212, −4.73008471518844336383240396338, −4.04460081770474790060966716943, −2.84088758892071107493637263760, −0.39350901278907161403994640454,
0.39350901278907161403994640454, 2.84088758892071107493637263760, 4.04460081770474790060966716943, 4.73008471518844336383240396338, 5.44520935012477249235218645212, 6.68362021617558605519586398154, 7.52014255219375669390744872053, 8.552279691581675412021506433262, 8.930855992037741878883741687395, 10.33567284921376614507540719534
