| L(s) = 1 | + 1.03e13·2-s − 5.11e26·4-s − 2.06e31·5-s − 1.95e37·7-s − 1.17e40·8-s − 2.14e44·10-s + 3.33e46·11-s + 6.87e49·13-s − 2.02e50·14-s + 1.94e53·16-s + 2.38e54·17-s + 1.04e57·19-s + 1.05e58·20-s + 3.46e59·22-s + 1.76e60·23-s + 2.66e62·25-s + 7.14e62·26-s + 9.97e63·28-s − 3.58e64·29-s − 5.07e65·31-s + 9.28e66·32-s + 2.47e67·34-s + 4.03e68·35-s + 3.65e69·37-s + 1.08e70·38-s + 2.42e71·40-s + 8.30e71·41-s + ⋯ |
| L(s) = 1 | + 0.417·2-s − 0.825·4-s − 1.62·5-s − 0.482·7-s − 0.762·8-s − 0.679·10-s + 1.51·11-s + 1.84·13-s − 0.201·14-s + 0.507·16-s + 0.419·17-s + 1.29·19-s + 1.34·20-s + 0.634·22-s + 0.447·23-s + 1.64·25-s + 0.772·26-s + 0.398·28-s − 0.300·29-s − 0.218·31-s + 0.974·32-s + 0.175·34-s + 0.785·35-s + 0.600·37-s + 0.541·38-s + 1.24·40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(90-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+89/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(45)\) |
\(\approx\) |
\(2.393049022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393049022\) |
| \(L(\frac{91}{2})\) |
|
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 | \( 1 - 1.03e13T + 6.18e26T^{2} \) |
| 5 | \( 1 + 2.06e31T + 1.61e62T^{2} \) |
| 7 | \( 1 + 1.95e37T + 1.63e75T^{2} \) |
| 11 | \( 1 - 3.33e46T + 4.83e92T^{2} \) |
| 13 | \( 1 - 6.87e49T + 1.38e99T^{2} \) |
| 17 | \( 1 - 2.38e54T + 3.23e109T^{2} \) |
| 19 | \( 1 - 1.04e57T + 6.44e113T^{2} \) |
| 23 | \( 1 - 1.76e60T + 1.56e121T^{2} \) |
| 29 | \( 1 + 3.58e64T + 1.42e130T^{2} \) |
| 31 | \( 1 + 5.07e65T + 5.38e132T^{2} \) |
| 37 | \( 1 - 3.65e69T + 3.71e139T^{2} \) |
| 41 | \( 1 - 8.30e71T + 3.44e143T^{2} \) |
| 43 | \( 1 - 1.00e72T + 2.39e145T^{2} \) |
| 47 | \( 1 - 1.68e74T + 6.55e148T^{2} \) |
| 53 | \( 1 - 3.74e76T + 2.88e153T^{2} \) |
| 59 | \( 1 - 6.54e78T + 4.03e157T^{2} \) |
| 61 | \( 1 - 6.58e78T + 7.84e158T^{2} \) |
| 67 | \( 1 + 6.71e80T + 3.31e162T^{2} \) |
| 71 | \( 1 - 3.76e82T + 5.78e164T^{2} \) |
| 73 | \( 1 - 8.64e81T + 6.85e165T^{2} \) |
| 79 | \( 1 - 1.41e84T + 7.74e168T^{2} \) |
| 83 | \( 1 - 6.96e84T + 6.27e170T^{2} \) |
| 89 | \( 1 - 1.63e86T + 3.13e173T^{2} \) |
| 97 | \( 1 + 3.58e88T + 6.64e176T^{2} \) |
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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
−9.464085247916401918923016427165, −8.735716553842526917071674735493, −7.78041764102194877801035360180, −6.60824839300947341506919070178, −5.57000400430171529767868352426, −4.22385759946225013625452970169, −3.74583904965024805848748802286, −3.20152327864442454637739936652, −1.06396653972717636088716674742, −0.69618780820461278742586227644,
0.69618780820461278742586227644, 1.06396653972717636088716674742, 3.20152327864442454637739936652, 3.74583904965024805848748802286, 4.22385759946225013625452970169, 5.57000400430171529767868352426, 6.60824839300947341506919070178, 7.78041764102194877801035360180, 8.735716553842526917071674735493, 9.464085247916401918923016427165
