| L(s) = 1 | − 2·2-s + 2·4-s + 2·7-s + 5·11-s − 4·14-s − 4·16-s + 4·17-s − 19-s − 10·22-s + 4·28-s − 2·29-s + 7·31-s + 8·32-s − 8·34-s + 10·37-s + 2·38-s + 41-s − 10·43-s + 10·44-s − 6·47-s − 3·49-s − 6·53-s + 4·58-s + 9·61-s − 14·62-s − 8·64-s − 14·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.755·7-s + 1.50·11-s − 1.06·14-s − 16-s + 0.970·17-s − 0.229·19-s − 2.13·22-s + 0.755·28-s − 0.371·29-s + 1.25·31-s + 1.41·32-s − 1.37·34-s + 1.64·37-s + 0.324·38-s + 0.156·41-s − 1.52·43-s + 1.50·44-s − 0.875·47-s − 3/7·49-s − 0.824·53-s + 0.525·58-s + 1.15·61-s − 1.77·62-s − 64-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.00313701643802, −13.20729807831219, −12.96183448528997, −11.91888199590763, −11.76250400670475, −11.44546721043773, −10.79643237518221, −10.28693716239374, −9.757693999808608, −9.500914557390860, −8.884734334838591, −8.440719407649417, −7.988608370910131, −7.620052569516583, −6.981905911304261, −6.438872437577524, −6.056318013285575, −5.212717995829082, −4.543232676737131, −4.227003031277600, −3.388310568864357, −2.742766781505085, −1.860081661898174, −1.375699863914392, −0.9906798746450057, 0,
0.9906798746450057, 1.375699863914392, 1.860081661898174, 2.742766781505085, 3.388310568864357, 4.227003031277600, 4.543232676737131, 5.212717995829082, 6.056318013285575, 6.438872437577524, 6.981905911304261, 7.620052569516583, 7.988608370910131, 8.440719407649417, 8.884734334838591, 9.500914557390860, 9.757693999808608, 10.28693716239374, 10.79643237518221, 11.44546721043773, 11.76250400670475, 11.91888199590763, 12.96183448528997, 13.20729807831219, 14.00313701643802
