| L(s) = 1 | + 2-s − 3-s + 4-s − 2.34·5-s − 6-s + 3.57·7-s + 8-s + 9-s − 2.34·10-s + 2.71·11-s − 12-s + 5.41·13-s + 3.57·14-s + 2.34·15-s + 16-s − 3.87·17-s + 18-s − 2.34·20-s − 3.57·21-s + 2.71·22-s − 8.23·23-s − 24-s + 0.509·25-s + 5.41·26-s − 27-s + 3.57·28-s + 3.53·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.04·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 0.333·9-s − 0.742·10-s + 0.819·11-s − 0.288·12-s + 1.50·13-s + 0.955·14-s + 0.606·15-s + 0.250·16-s − 0.940·17-s + 0.235·18-s − 0.524·20-s − 0.779·21-s + 0.579·22-s − 1.71·23-s − 0.204·24-s + 0.101·25-s + 1.06·26-s − 0.192·27-s + 0.675·28-s + 0.655·29-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
| \(L(1)\) |
\(\approx\) |
\(2.441179223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.441179223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 8.30T + 53T^{2} \) |
| 59 | \( 1 - 2.73T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 8.17T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.740794666247843875898366627111, −8.234802166586672048974903847153, −7.51344458179378749337477746297, −6.51141979290713474130333236007, −5.97004086615526249419146877733, −4.87839641419007151221110633635, −4.15488237515851959146740096688, −3.76794019526435460596299440498, −2.13750674071357203653735324162, −1.01566411363640445936628896811,
1.01566411363640445936628896811, 2.13750674071357203653735324162, 3.76794019526435460596299440498, 4.15488237515851959146740096688, 4.87839641419007151221110633635, 5.97004086615526249419146877733, 6.51141979290713474130333236007, 7.51344458179378749337477746297, 8.234802166586672048974903847153, 8.740794666247843875898366627111
