| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4.07·7-s − 8-s + 9-s − 10-s + 12-s − 4.13·13-s + 4.07·14-s + 15-s + 16-s + 5.97·17-s − 18-s − 5.35·19-s + 20-s − 4.07·21-s + 3.33·23-s − 24-s + 25-s + 4.13·26-s + 27-s − 4.07·28-s + 2.45·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.53·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.14·13-s + 1.08·14-s + 0.258·15-s + 0.250·16-s + 1.44·17-s − 0.235·18-s − 1.22·19-s + 0.223·20-s − 0.888·21-s + 0.695·23-s − 0.204·24-s + 0.200·25-s + 0.811·26-s + 0.192·27-s − 0.769·28-s + 0.455·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.305129864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305129864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4.97T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 8.43T + 97T^{2} \) |
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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
−8.786227056791963485793671211979, −7.68699458427271331266412795075, −7.29614389059475380143768472998, −6.37856459335755629157000476959, −5.85637220447766366951203618735, −4.71916645637676499641067974246, −3.56139946786399743164250102325, −2.86157182761029892584425456922, −2.11934066981889291897934828967, −0.69806775574472982221494687111,
0.69806775574472982221494687111, 2.11934066981889291897934828967, 2.86157182761029892584425456922, 3.56139946786399743164250102325, 4.71916645637676499641067974246, 5.85637220447766366951203618735, 6.37856459335755629157000476959, 7.29614389059475380143768472998, 7.68699458427271331266412795075, 8.786227056791963485793671211979
