L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s − 4·13-s − 2·15-s + 17-s − 8·19-s − 4·21-s − 25-s − 27-s + 10·31-s + 8·35-s − 8·37-s + 4·39-s + 10·41-s − 8·43-s + 2·45-s + 10·47-s + 9·49-s − 51-s + 12·53-s + 8·57-s + 8·59-s − 2·61-s + 4·63-s − 8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 0.242·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.79·31-s + 1.35·35-s − 1.31·37-s + 0.640·39-s + 1.56·41-s − 1.21·43-s + 0.298·45-s + 1.45·47-s + 9/7·49-s − 0.140·51-s + 1.64·53-s + 1.05·57-s + 1.04·59-s − 0.256·61-s + 0.503·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.60235194059869, −12.06009327722039, −11.84176302332632, −11.39522734960297, −10.73024493330498, −10.41042565199746, −10.18257117005643, −9.623004952868779, −9.005933396648362, −8.530348373561420, −8.208647160883161, −7.643577502267691, −7.056791670371974, −6.776132771698672, −6.074248077682699, −5.611715133331400, −5.368514703287414, −4.672467997200503, −4.351913418158305, −4.018719699484635, −2.902046300853925, −2.406893428179430, −1.991624449581565, −1.482220331043264, −0.8184830006236192, 0,
0.8184830006236192, 1.482220331043264, 1.991624449581565, 2.406893428179430, 2.902046300853925, 4.018719699484635, 4.351913418158305, 4.672467997200503, 5.368514703287414, 5.611715133331400, 6.074248077682699, 6.776132771698672, 7.056791670371974, 7.643577502267691, 8.208647160883161, 8.530348373561420, 9.005933396648362, 9.623004952868779, 10.18257117005643, 10.41042565199746, 10.73024493330498, 11.39522734960297, 11.84176302332632, 12.06009327722039, 12.60235194059869