| L(s) = 1 | + 2-s − 3-s + 4-s − 3.53·5-s − 6-s + 3.71·7-s + 8-s + 9-s − 3.53·10-s − 5.29·11-s − 12-s − 0.226·13-s + 3.71·14-s + 3.53·15-s + 16-s − 1.65·17-s + 18-s − 3.53·20-s − 3.71·21-s − 5.29·22-s + 8.68·23-s − 24-s + 7.47·25-s − 0.226·26-s − 27-s + 3.71·28-s + 0.120·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.57·5-s − 0.408·6-s + 1.40·7-s + 0.353·8-s + 0.333·9-s − 1.11·10-s − 1.59·11-s − 0.288·12-s − 0.0628·13-s + 0.993·14-s + 0.911·15-s + 0.250·16-s − 0.400·17-s + 0.235·18-s − 0.789·20-s − 0.811·21-s − 1.12·22-s + 1.80·23-s − 0.204·24-s + 1.49·25-s − 0.0444·26-s − 0.192·27-s + 0.702·28-s + 0.0223·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752708725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752708725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.53T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 0.226T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 0.120T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 + 2.50T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 - 0.218T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 - 1.92T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.754799930959233686664831858591, −8.007163940479088417142477848001, −7.56590854427988486384970129250, −6.86242043576251519762161699678, −5.61710732190131596883591199080, −4.74482983418777640433021976555, −4.63106896840390127342649969178, −3.41430015207935316398647035398, −2.38004122342987678282184697171, −0.804558554932805246678121583302,
0.804558554932805246678121583302, 2.38004122342987678282184697171, 3.41430015207935316398647035398, 4.63106896840390127342649969178, 4.74482983418777640433021976555, 5.61710732190131596883591199080, 6.86242043576251519762161699678, 7.56590854427988486384970129250, 8.007163940479088417142477848001, 8.754799930959233686664831858591
