| L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s − 5-s + 1.73·6-s − 7-s − 1.73·8-s + 9-s − 1.73·10-s + 2.73·11-s + 0.999·12-s − 4.73·13-s − 1.73·14-s − 15-s − 5·16-s − 7.46·17-s + 1.73·18-s − 3.46·19-s − 0.999·20-s − 21-s + 4.73·22-s + 23-s − 1.73·24-s + 25-s − 8.19·26-s + 27-s − 0.999·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s − 0.377·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.823·11-s + 0.288·12-s − 1.31·13-s − 0.462·14-s − 0.258·15-s − 1.25·16-s − 1.81·17-s + 0.408·18-s − 0.794·19-s − 0.223·20-s − 0.218·21-s + 1.00·22-s + 0.208·23-s − 0.353·24-s + 0.200·25-s − 1.60·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 - 0.732T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 3.66T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 6.53T + 89T^{2} \) |
| 97 | \( 1 - 2.92T + 97T^{2} \) |
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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
−8.828646491065890938857464803399, −7.64316451319783038898097192388, −6.76510664725334241484654105232, −6.35667403943400842252262990263, −5.10800366197472759380565668438, −4.41922006392416609691257372630, −3.87249194634555114084418808383, −2.89506948493770158158820331283, −2.10049246126520267332129729304, 0,
2.10049246126520267332129729304, 2.89506948493770158158820331283, 3.87249194634555114084418808383, 4.41922006392416609691257372630, 5.10800366197472759380565668438, 6.35667403943400842252262990263, 6.76510664725334241484654105232, 7.64316451319783038898097192388, 8.828646491065890938857464803399
