| L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 4·11-s − 3·13-s + 3·17-s + 3·21-s + 23-s − 5·25-s + 5·27-s + 29-s + 4·31-s − 4·33-s + 6·37-s + 3·39-s + 4·41-s − 4·47-s + 2·49-s − 3·51-s + 53-s − 9·59-s + 8·61-s + 6·63-s + 3·67-s − 69-s − 12·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 1.20·11-s − 0.832·13-s + 0.727·17-s + 0.654·21-s + 0.208·23-s − 25-s + 0.962·27-s + 0.185·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.480·39-s + 0.624·41-s − 0.583·47-s + 2/7·49-s − 0.420·51-s + 0.137·53-s − 1.17·59-s + 1.02·61-s + 0.755·63-s + 0.366·67-s − 0.120·69-s − 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.20178020335539, −13.52298543368458, −13.07677855355200, −12.48397947488753, −12.05818276616724, −11.70313700684164, −11.30650654836681, −10.59334779987474, −10.04779185697139, −9.580518679716231, −9.336779637665911, −8.609105898196868, −8.073248299533874, −7.425823816724080, −6.853468158881925, −6.418336322953350, −5.843938153260963, −5.633222091508885, −4.688983953868366, −4.285601052689922, −3.508730462153724, −3.010453638779015, −2.467214939123245, −1.499201910419678, −0.7353777457843679, 0,
0.7353777457843679, 1.499201910419678, 2.467214939123245, 3.010453638779015, 3.508730462153724, 4.285601052689922, 4.688983953868366, 5.633222091508885, 5.843938153260963, 6.418336322953350, 6.853468158881925, 7.425823816724080, 8.073248299533874, 8.609105898196868, 9.336779637665911, 9.580518679716231, 10.04779185697139, 10.59334779987474, 11.30650654836681, 11.70313700684164, 12.05818276616724, 12.48397947488753, 13.07677855355200, 13.52298543368458, 14.20178020335539
