| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s − 11-s − 6·13-s + 2·15-s + 6·17-s − 4·21-s − 25-s − 27-s + 6·29-s + 33-s − 8·35-s − 6·37-s + 6·39-s + 10·41-s − 8·43-s − 2·45-s + 9·49-s − 6·51-s − 6·53-s + 2·55-s − 4·59-s − 2·61-s + 4·63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s + 1.45·17-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s − 1.35·35-s − 0.986·37-s + 0.960·39-s + 1.56·41-s − 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 0.520·59-s − 0.256·61-s + 0.503·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.10886973908366, −13.79760202928811, −12.72802279498133, −12.48176690592417, −11.94716826030578, −11.75784379563366, −11.16355090503009, −10.71717761043013, −10.14237551948811, −9.749199126283078, −9.103602067783529, −8.210664648082454, −7.987253109376025, −7.660200126193901, −7.089629980480614, −6.549339099569952, −5.620847676504438, −5.246588994396682, −4.807889756084496, −4.373905823860254, −3.689074337331924, −2.957862955129801, −2.259179922103932, −1.536211843727237, −0.8213798610151977, 0,
0.8213798610151977, 1.536211843727237, 2.259179922103932, 2.957862955129801, 3.689074337331924, 4.373905823860254, 4.807889756084496, 5.246588994396682, 5.620847676504438, 6.549339099569952, 7.089629980480614, 7.660200126193901, 7.987253109376025, 8.210664648082454, 9.103602067783529, 9.749199126283078, 10.14237551948811, 10.71717761043013, 11.16355090503009, 11.75784379563366, 11.94716826030578, 12.48176690592417, 12.72802279498133, 13.79760202928811, 14.10886973908366
