| L(s) = 1 | − 2-s + 3-s + 4-s + 3.27·5-s − 6-s − 7-s − 8-s + 9-s − 3.27·10-s + 4·11-s + 12-s + 14-s + 3.27·15-s + 16-s + 3·17-s − 18-s + 8.54·19-s + 3.27·20-s − 21-s − 4·22-s + 2.27·23-s − 24-s + 5.72·25-s + 27-s − 28-s + 0.725·29-s − 3.27·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.46·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 1.20·11-s + 0.288·12-s + 0.267·14-s + 0.845·15-s + 0.250·16-s + 0.727·17-s − 0.235·18-s + 1.96·19-s + 0.732·20-s − 0.218·21-s − 0.852·22-s + 0.474·23-s − 0.204·24-s + 1.14·25-s + 0.192·27-s − 0.188·28-s + 0.134·29-s − 0.597·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.025916910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025916910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 8.54T + 19T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 7.27T + 37T^{2} \) |
| 41 | \( 1 + 0.725T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 8.54T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 9T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 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
−7.942793483557331915388758580715, −7.24902609699782249377691633453, −6.70032907335442660600515889677, −5.81438657650852241095066224780, −5.47556775664803999647498203844, −4.22972319632168507443973899713, −3.23303425284918098769209398022, −2.68038826018460051733773395230, −1.57543823826712693795743274494, −1.09358988712774501287963897065,
1.09358988712774501287963897065, 1.57543823826712693795743274494, 2.68038826018460051733773395230, 3.23303425284918098769209398022, 4.22972319632168507443973899713, 5.47556775664803999647498203844, 5.81438657650852241095066224780, 6.70032907335442660600515889677, 7.24902609699782249377691633453, 7.942793483557331915388758580715
