| L(s) = 1 | + 3.25·3-s − 3.25·5-s + 7.62·9-s + 11-s + 1.22·13-s − 10.6·15-s − 6.51·17-s + 5.29·19-s + 6.62·23-s + 5.62·25-s + 15.0·27-s + 6·29-s − 8.55·31-s + 3.25·33-s − 4.62·37-s + 4·39-s + 6.51·41-s + 4·43-s − 24.8·45-s − 4.06·47-s − 21.2·51-s + 6·53-s − 3.25·55-s + 17.2·57-s + 13.8·59-s − 5.29·61-s − 4·65-s + ⋯ |
| L(s) = 1 | + 1.88·3-s − 1.45·5-s + 2.54·9-s + 0.301·11-s + 0.340·13-s − 2.74·15-s − 1.58·17-s + 1.21·19-s + 1.38·23-s + 1.12·25-s + 2.90·27-s + 1.11·29-s − 1.53·31-s + 0.567·33-s − 0.760·37-s + 0.640·39-s + 1.01·41-s + 0.609·43-s − 3.70·45-s − 0.592·47-s − 2.97·51-s + 0.824·53-s − 0.439·55-s + 2.28·57-s + 1.80·59-s − 0.677·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.261658438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261658438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 5.29T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 8.97T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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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
−8.429755029366641891972883490510, −7.72116520922986993952618213894, −7.19919309028618499265243739977, −6.67088580837878569488890252335, −5.06944691411358868197615624839, −4.26637463420865330525984251357, −3.68272321712631203172830760905, −3.09022195171897738199003612181, −2.20182851038987176416768475373, −0.964251217982307487985884846341,
0.964251217982307487985884846341, 2.20182851038987176416768475373, 3.09022195171897738199003612181, 3.68272321712631203172830760905, 4.26637463420865330525984251357, 5.06944691411358868197615624839, 6.67088580837878569488890252335, 7.19919309028618499265243739977, 7.72116520922986993952618213894, 8.429755029366641891972883490510
