| L(s) = 1 | + 2.04e9·2-s + 1.88e18·4-s + 2.27e20·5-s − 4.42e25·7-s − 8.53e26·8-s + 4.66e29·10-s − 7.45e31·11-s + 1.57e34·13-s − 9.06e34·14-s − 6.10e36·16-s + 3.16e37·17-s + 4.06e38·19-s + 4.30e38·20-s − 1.52e41·22-s − 5.44e40·23-s − 4.28e42·25-s + 3.23e43·26-s − 8.35e43·28-s − 2.13e44·29-s − 2.29e45·31-s − 1.05e46·32-s + 6.48e46·34-s − 1.00e46·35-s − 1.07e48·37-s + 8.32e47·38-s − 1.94e47·40-s + 1.40e49·41-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.819·4-s + 0.109·5-s − 0.742·7-s − 0.243·8-s + 0.147·10-s − 1.28·11-s + 1.67·13-s − 1.00·14-s − 1.14·16-s + 0.936·17-s + 0.404·19-s + 0.0896·20-s − 1.73·22-s − 0.159·23-s − 0.988·25-s + 2.25·26-s − 0.607·28-s − 0.532·29-s − 0.750·31-s − 1.30·32-s + 1.26·34-s − 0.0812·35-s − 1.58·37-s + 0.545·38-s − 0.0266·40-s + 0.905·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(62-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+61/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(31)\) |
\(\approx\) |
\(3.743991479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743991479\) |
| \(L(\frac{63}{2})\) |
|
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 | \( 1 - 2.04e9T + 2.30e18T^{2} \) |
| 5 | \( 1 - 2.27e20T + 4.33e42T^{2} \) |
| 7 | \( 1 + 4.42e25T + 3.55e51T^{2} \) |
| 11 | \( 1 + 7.45e31T + 3.34e63T^{2} \) |
| 13 | \( 1 - 1.57e34T + 8.92e67T^{2} \) |
| 17 | \( 1 - 3.16e37T + 1.14e75T^{2} \) |
| 19 | \( 1 - 4.06e38T + 1.00e78T^{2} \) |
| 23 | \( 1 + 5.44e40T + 1.16e83T^{2} \) |
| 29 | \( 1 + 2.13e44T + 1.60e89T^{2} \) |
| 31 | \( 1 + 2.29e45T + 9.39e90T^{2} \) |
| 37 | \( 1 + 1.07e48T + 4.57e95T^{2} \) |
| 41 | \( 1 - 1.40e49T + 2.39e98T^{2} \) |
| 43 | \( 1 - 5.79e49T + 4.38e99T^{2} \) |
| 47 | \( 1 - 1.51e51T + 9.95e101T^{2} \) |
| 53 | \( 1 - 4.58e52T + 1.51e105T^{2} \) |
| 59 | \( 1 - 1.78e54T + 1.05e108T^{2} \) |
| 61 | \( 1 - 4.41e54T + 8.03e108T^{2} \) |
| 67 | \( 1 - 2.75e55T + 2.45e111T^{2} \) |
| 71 | \( 1 + 1.08e56T + 8.44e112T^{2} \) |
| 73 | \( 1 + 1.85e56T + 4.59e113T^{2} \) |
| 79 | \( 1 + 4.07e57T + 5.69e115T^{2} \) |
| 83 | \( 1 + 2.46e58T + 1.15e117T^{2} \) |
| 89 | \( 1 - 6.63e57T + 8.18e118T^{2} \) |
| 97 | \( 1 + 2.67e60T + 1.55e121T^{2} \) |
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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
−11.34005248550990008778494823362, −10.10241588282450270522395388269, −8.714845780826765495910265978508, −7.27469057199787256797714827546, −5.86235900943320710475422906455, −5.48546808147113041753025652739, −3.92332728743866464032787986881, −3.31921361865746127951304442588, −2.18722410172206447130693076617, −0.63034467276491521921525017822,
0.63034467276491521921525017822, 2.18722410172206447130693076617, 3.31921361865746127951304442588, 3.92332728743866464032787986881, 5.48546808147113041753025652739, 5.86235900943320710475422906455, 7.27469057199787256797714827546, 8.714845780826765495910265978508, 10.10241588282450270522395388269, 11.34005248550990008778494823362
