| L(s) = 1 | + 3·2-s + 8·3-s + 4-s + 24·6-s + 28·7-s − 21·8-s + 37·9-s − 24·11-s + 8·12-s + 58·13-s + 84·14-s − 71·16-s − 17·17-s + 111·18-s + 116·19-s + 224·21-s − 72·22-s + 60·23-s − 168·24-s + 174·26-s + 80·27-s + 28·28-s + 30·29-s − 172·31-s − 45·32-s − 192·33-s − 51·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.53·3-s + 1/8·4-s + 1.63·6-s + 1.51·7-s − 0.928·8-s + 1.37·9-s − 0.657·11-s + 0.192·12-s + 1.23·13-s + 1.60·14-s − 1.10·16-s − 0.242·17-s + 1.45·18-s + 1.40·19-s + 2.32·21-s − 0.697·22-s + 0.543·23-s − 1.42·24-s + 1.31·26-s + 0.570·27-s + 0.188·28-s + 0.192·29-s − 0.996·31-s − 0.248·32-s − 1.01·33-s − 0.257·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.829350163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.829350163\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 342 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 484 T + p^{3} T^{2} \) |
| 83 | \( 1 + 756 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−10.98915617154887282550038301050, −9.635514101299671274909123796034, −8.679411753169250478902992304805, −8.202174913113442602794174208815, −7.20546262771878513899161035458, −5.62345556355855212309398416718, −4.76020928664743998422692575287, −3.72576202998137396834861995440, −2.86669902994824236723004758348, −1.55363240263121170151395803281,
1.55363240263121170151395803281, 2.86669902994824236723004758348, 3.72576202998137396834861995440, 4.76020928664743998422692575287, 5.62345556355855212309398416718, 7.20546262771878513899161035458, 8.202174913113442602794174208815, 8.679411753169250478902992304805, 9.635514101299671274909123796034, 10.98915617154887282550038301050
