| L(s) = 1 | + (−1.53 + 1.28i)2-s + (−3.05 − 1.11i)3-s + (0.694 − 3.93i)4-s + (−1.91 − 10.8i)5-s + (6.11 − 2.22i)6-s + (9.80 − 16.9i)7-s + (4.00 + 6.92i)8-s + (−12.5 − 10.5i)9-s + (16.9 + 14.2i)10-s + (−6.14 − 10.6i)11-s + (−6.51 + 11.2i)12-s + (−6.72 + 2.44i)13-s + (6.81 + 38.6i)14-s + (−6.24 + 35.4i)15-s + (−15.0 − 5.47i)16-s + (−69.7 + 58.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.588 − 0.214i)3-s + (0.0868 − 0.492i)4-s + (−0.171 − 0.973i)5-s + (0.416 − 0.151i)6-s + (0.529 − 0.917i)7-s + (0.176 + 0.306i)8-s + (−0.465 − 0.390i)9-s + (0.535 + 0.449i)10-s + (−0.168 − 0.291i)11-s + (−0.156 + 0.271i)12-s + (−0.143 + 0.0522i)13-s + (0.130 + 0.737i)14-s + (−0.107 + 0.610i)15-s + (−0.234 − 0.0855i)16-s + (−0.994 + 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.508140 - 0.457842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508140 - 0.457842i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.53 - 1.28i)T \) |
| 19 | \( 1 + (-82.6 - 4.58i)T \) |
| good | 3 | \( 1 + (3.05 + 1.11i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (1.91 + 10.8i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-9.80 + 16.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (6.14 + 10.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (6.72 - 2.44i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (69.7 - 58.4i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-16.6 + 94.6i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-34.5 - 28.9i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-146. + 253. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (307. + 111. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-40.5 - 230. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-242. - 203. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (92.6 - 525. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-226. + 190. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-147. + 837. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-246. - 206. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (91.0 + 516. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (900. + 327. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (428. + 155. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-567. + 982. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.31e3 + 479. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-391. + 328. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−15.88245353895699804014589555109, −14.48004853330860632141955793318, −13.15729822469865423935642881301, −11.76835276626537178299739623077, −10.64610562843830002606203990693, −9.007686181641993919022724484302, −7.84655097350168315435724449925, −6.26092387543254442774072864338, −4.67461708614645900843130649577, −0.76530266602652360129942436482,
2.68927704843659707448108948956, 5.20057269558365043844391298834, 7.05900730118505090731163474279, 8.610186900702692305749652922380, 10.13561491718765315630927127790, 11.32074701015720733993249974252, 11.86614287011851162159996755281, 13.72972272849270664641580318526, 15.12127012625949659685890011140, 16.12525910096494866759916635446
