| L(s) = 1 | + 3.61e10·2-s − 1.05e21·4-s − 4.57e24·5-s − 2.20e29·7-s − 1.23e32·8-s − 1.65e35·10-s − 4.33e36·11-s + 3.74e39·13-s − 7.94e39·14-s − 1.96e42·16-s − 4.48e43·17-s + 3.60e45·19-s + 4.83e45·20-s − 1.56e47·22-s − 2.49e48·23-s − 2.14e49·25-s + 1.35e50·26-s + 2.32e50·28-s + 4.79e51·29-s + 7.02e52·31-s + 2.20e53·32-s − 1.61e54·34-s + 1.00e54·35-s + 1.51e55·37-s + 1.30e56·38-s + 5.64e56·40-s − 3.49e56·41-s + ⋯ |
| L(s) = 1 | + 0.743·2-s − 0.447·4-s − 0.703·5-s − 0.219·7-s − 1.07·8-s − 0.522·10-s − 0.464·11-s + 1.06·13-s − 0.163·14-s − 0.351·16-s − 0.934·17-s + 1.44·19-s + 0.314·20-s − 0.345·22-s − 1.13·23-s − 0.505·25-s + 0.793·26-s + 0.0982·28-s + 0.582·29-s + 0.800·31-s + 0.814·32-s − 0.694·34-s + 0.154·35-s + 0.322·37-s + 1.07·38-s + 0.756·40-s − 0.194·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3.61e10T + 2.36e21T^{2} \) |
| 5 | \( 1 + 4.57e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 2.20e29T + 1.00e60T^{2} \) |
| 11 | \( 1 + 4.33e36T + 8.68e73T^{2} \) |
| 13 | \( 1 - 3.74e39T + 1.23e79T^{2} \) |
| 17 | \( 1 + 4.48e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 3.60e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 2.49e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 4.79e51T + 6.76e103T^{2} \) |
| 31 | \( 1 - 7.02e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 1.51e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 3.49e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 8.05e57T + 9.46e115T^{2} \) |
| 47 | \( 1 - 2.12e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 2.29e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 4.78e61T + 5.37e125T^{2} \) |
| 61 | \( 1 + 3.88e62T + 5.73e126T^{2} \) |
| 67 | \( 1 - 8.84e64T + 4.48e129T^{2} \) |
| 71 | \( 1 - 2.94e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 1.51e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 2.53e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 7.66e67T + 1.79e136T^{2} \) |
| 89 | \( 1 + 1.19e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 1.55e70T + 1.15e141T^{2} \) |
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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
−9.928831014176852920194072567918, −8.748842031718046983664781136947, −7.80296482161629265952872602644, −6.38133689150944833472102819617, −5.41916456529384230358101441132, −4.28198679929607729785242181218, −3.61925345096503100427148864700, −2.56762767775414945136647072516, −0.951793704272535803111443817681, 0,
0.951793704272535803111443817681, 2.56762767775414945136647072516, 3.61925345096503100427148864700, 4.28198679929607729785242181218, 5.41916456529384230358101441132, 6.38133689150944833472102819617, 7.80296482161629265952872602644, 8.748842031718046983664781136947, 9.928831014176852920194072567918
