| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s + 4·14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s − 24-s + 25-s + 2·26-s − 27-s + 4·28-s + 6·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.359513070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.359513070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.30175509093558, −13.85963091804444, −13.62197747665921, −12.85684376822419, −12.15772849115649, −11.95322123453312, −11.45883909693719, −10.97293613990074, −10.31339557709329, −10.05059985148236, −9.341737973443094, −8.396324625445847, −8.176363400816943, −7.542667296108351, −6.933107218597967, −6.292594888535864, −5.776966035779799, −5.140767037115811, −4.948518269033866, −4.275468408972610, −3.491184913195611, −2.897073090378688, −2.027393466319607, −1.287033357223867, −0.9384267492322601,
0.9384267492322601, 1.287033357223867, 2.027393466319607, 2.897073090378688, 3.491184913195611, 4.275468408972610, 4.948518269033866, 5.140767037115811, 5.776966035779799, 6.292594888535864, 6.933107218597967, 7.542667296108351, 8.176363400816943, 8.396324625445847, 9.341737973443094, 10.05059985148236, 10.31339557709329, 10.97293613990074, 11.45883909693719, 11.95322123453312, 12.15772849115649, 12.85684376822419, 13.62197747665921, 13.85963091804444, 14.30175509093558
