| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 11-s + 2·13-s + 2·14-s − 16-s − 2·17-s − 6·19-s − 22-s − 4·23-s + 2·26-s − 2·28-s − 6·29-s + 4·31-s + 5·32-s − 2·34-s + 6·37-s − 6·38-s − 10·41-s − 6·43-s + 44-s − 4·46-s + 8·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.213·22-s − 0.834·23-s + 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.986·37-s − 0.973·38-s − 1.56·41-s − 0.914·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.426508541426244909616098012576, −8.037404731553225089340453872858, −6.81603726027801660997015654358, −6.02983209069794989603640383676, −5.32946274439723089843787755675, −4.42806580035514350183447341926, −3.98159649274510364542505380476, −2.82049928275954817840186820477, −1.70002773304928331393197381104, 0,
1.70002773304928331393197381104, 2.82049928275954817840186820477, 3.98159649274510364542505380476, 4.42806580035514350183447341926, 5.32946274439723089843787755675, 6.02983209069794989603640383676, 6.81603726027801660997015654358, 8.037404731553225089340453872858, 8.426508541426244909616098012576
