| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s + 25-s + 27-s + 28-s + 9·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.144842188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144842188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 349 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.25140132303788, −15.91555181726849, −15.42527579974137, −14.72138105471962, −14.18508709122182, −13.73739911295227, −13.28894970927238, −12.74669842720010, −11.79439857202693, −11.68597990567819, −10.69511596184891, −10.07322522551455, −9.729577133498347, −8.720925617950207, −8.235876937092878, −7.616880145610547, −6.836057718900128, −6.301911574684966, −5.396254886178816, −4.912478026629144, −4.202413336871992, −3.322835997379591, −2.635959717588869, −1.991919750467729, −0.9494431811259593,
0.9494431811259593, 1.991919750467729, 2.635959717588869, 3.322835997379591, 4.202413336871992, 4.912478026629144, 5.396254886178816, 6.301911574684966, 6.836057718900128, 7.616880145610547, 8.235876937092878, 8.720925617950207, 9.729577133498347, 10.07322522551455, 10.69511596184891, 11.68597990567819, 11.79439857202693, 12.74669842720010, 13.28894970927238, 13.73739911295227, 14.18508709122182, 14.72138105471962, 15.42527579974137, 15.91555181726849, 16.25140132303788
