| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3.23·7-s − 8-s + 9-s − 10-s − 12-s − 0.618·13-s − 3.23·14-s − 15-s + 16-s − 2.85·17-s − 18-s − 5.23·19-s + 20-s − 3.23·21-s + 3.61·23-s + 24-s + 25-s + 0.618·26-s − 27-s + 3.23·28-s + 5.09·29-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 + 1.22·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s − 0.171·13-s − 0.864·14-s − 0.258·15-s + 0.250·16-s − 0.692·17-s − 0.235·18-s − 1.20·19-s + 0.223·20-s − 0.706·21-s + 0.754·23-s + 0.204·24-s + 0.200·25-s + 0.121·26-s − 0.192·27-s + 0.611·28-s + 0.945·29-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
| \(L(1)\) |
\(\approx\) |
\(1.334517320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334517320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 0.854T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.854T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 6.85T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.32T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.665926877055264178417364411515, −7.82633832914397322403359288206, −7.12733170949391503150061085537, −6.38229205961320694963924402263, −5.62838606568684086816012750006, −4.80654937107170351980735626848, −4.11815855168949038905909172026, −2.59313387176211811118089270595, −1.85202457403829510734146406584, −0.792261685069056444484626206977,
0.792261685069056444484626206977, 1.85202457403829510734146406584, 2.59313387176211811118089270595, 4.11815855168949038905909172026, 4.80654937107170351980735626848, 5.62838606568684086816012750006, 6.38229205961320694963924402263, 7.12733170949391503150061085537, 7.82633832914397322403359288206, 8.665926877055264178417364411515
