| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 2·11-s − 5·13-s + 4·19-s + 3·21-s + 23-s − 9·27-s − 3·29-s − 5·31-s + 6·33-s − 4·37-s + 15·39-s + 5·41-s − 4·43-s − 11·47-s + 49-s − 12·57-s + 12·59-s − 6·61-s − 6·63-s − 16·67-s − 3·69-s − 7·71-s − 7·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 0.603·11-s − 1.38·13-s + 0.917·19-s + 0.654·21-s + 0.208·23-s − 1.73·27-s − 0.557·29-s − 0.898·31-s + 1.04·33-s − 0.657·37-s + 2.40·39-s + 0.780·41-s − 0.609·43-s − 1.60·47-s + 1/7·49-s − 1.58·57-s + 1.56·59-s − 0.768·61-s − 0.755·63-s − 1.95·67-s − 0.361·69-s − 0.830·71-s − 0.819·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.71209723892883, −14.40564247873593, −13.40331815486175, −13.14399666204728, −12.60142681572078, −12.16339286020473, −11.68404765744587, −11.32540124719775, −10.70778784354684, −10.20326853273899, −9.858625162139168, −9.355877254386506, −8.634778444359660, −7.743683005251122, −7.259516691739721, −7.025770914118463, −6.280577371336036, −5.629169577023324, −5.396559775259190, −4.772243314802946, −4.343116029911353, −3.413073281500564, −2.791588185081708, −1.875214771671731, −1.160675276299827, 0, 0,
1.160675276299827, 1.875214771671731, 2.791588185081708, 3.413073281500564, 4.343116029911353, 4.772243314802946, 5.396559775259190, 5.629169577023324, 6.280577371336036, 7.025770914118463, 7.259516691739721, 7.743683005251122, 8.634778444359660, 9.355877254386506, 9.858625162139168, 10.20326853273899, 10.70778784354684, 11.32540124719775, 11.68404765744587, 12.16339286020473, 12.60142681572078, 13.14399666204728, 13.40331815486175, 14.40564247873593, 14.71209723892883
