| L(s) = 1 | − 1.56·2-s + 0.452·4-s − 1.36·5-s + 3.35·7-s + 2.42·8-s + 2.13·10-s − 11-s + 6.49·13-s − 5.26·14-s − 4.70·16-s − 3.12·17-s − 0.766·19-s − 0.616·20-s + 1.56·22-s + 0.696·23-s − 3.14·25-s − 10.1·26-s + 1.52·28-s + 1.69·29-s − 0.344·31-s + 2.51·32-s + 4.90·34-s − 4.57·35-s + 5.23·37-s + 1.20·38-s − 3.29·40-s − 8.27·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.226·4-s − 0.608·5-s + 1.26·7-s + 0.856·8-s + 0.674·10-s − 0.301·11-s + 1.80·13-s − 1.40·14-s − 1.17·16-s − 0.758·17-s − 0.175·19-s − 0.137·20-s + 0.333·22-s + 0.145·23-s − 0.629·25-s − 1.99·26-s + 0.287·28-s + 0.314·29-s − 0.0618·31-s + 0.444·32-s + 0.840·34-s − 0.773·35-s + 0.861·37-s + 0.194·38-s − 0.521·40-s − 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 0.766T + 19T^{2} \) |
| 23 | \( 1 - 0.696T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 0.344T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 2.80T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 1.74T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + 5.21T + 97T^{2} \) |
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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
−8.014298460820076148332791664040, −7.36331538473436653197545877854, −6.48507930898308426940375802117, −5.63819898910745423405561023905, −4.56922743245857089784636526859, −4.27504052880962122136775350892, −3.18252417682436565581592539167, −1.83584478497526842878835898933, −1.26356533521483781037494196869, 0,
1.26356533521483781037494196869, 1.83584478497526842878835898933, 3.18252417682436565581592539167, 4.27504052880962122136775350892, 4.56922743245857089784636526859, 5.63819898910745423405561023905, 6.48507930898308426940375802117, 7.36331538473436653197545877854, 8.014298460820076148332791664040
