| L(s) = 1 | − 2-s − 0.198·3-s + 4-s − 0.890·5-s + 0.198·6-s + 7-s − 8-s − 2.96·9-s + 0.890·10-s − 0.664·11-s − 0.198·12-s − 14-s + 0.176·15-s + 16-s − 2.44·17-s + 2.96·18-s + 8.63·19-s − 0.890·20-s − 0.198·21-s + 0.664·22-s − 7.60·23-s + 0.198·24-s − 4.20·25-s + 1.18·27-s + 28-s + 3.60·29-s − 0.176·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.114·3-s + 0.5·4-s − 0.398·5-s + 0.0808·6-s + 0.377·7-s − 0.353·8-s − 0.986·9-s + 0.281·10-s − 0.200·11-s − 0.0571·12-s − 0.267·14-s + 0.0455·15-s + 0.250·16-s − 0.593·17-s + 0.697·18-s + 1.98·19-s − 0.199·20-s − 0.0432·21-s + 0.141·22-s − 1.58·23-s + 0.0404·24-s − 0.841·25-s + 0.227·27-s + 0.188·28-s + 0.669·29-s − 0.0321·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9057052945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9057052945\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.198T + 3T^{2} \) |
| 5 | \( 1 + 0.890T + 5T^{2} \) |
| 11 | \( 1 + 0.664T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 8.63T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 7.83T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 - 4.37T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.941497783467430875837762549869, −8.021369606565179782229790657747, −7.82903960399757875216656571291, −6.76666597037889926788497029203, −5.88188541896118568314861053403, −5.20291576196928205554149800304, −4.05191347239270475198756712866, −3.04501669062144818976111592416, −2.07897276430616856690517221055, −0.66749074865546196207883589210,
0.66749074865546196207883589210, 2.07897276430616856690517221055, 3.04501669062144818976111592416, 4.05191347239270475198756712866, 5.20291576196928205554149800304, 5.88188541896118568314861053403, 6.76666597037889926788497029203, 7.82903960399757875216656571291, 8.021369606565179782229790657747, 8.941497783467430875837762549869
