| L(s) = 1 | + (−2.26 + 5.18i)2-s − 10.8·3-s + (−21.7 − 23.5i)4-s + (24.5 − 56.0i)6-s + 163. i·7-s + (171. − 59.0i)8-s − 125.·9-s − 321. i·11-s + (234. + 254. i)12-s − 128.·13-s + (−848. − 371. i)14-s + (−82.2 + 1.02e3i)16-s − 2.11e3i·17-s + (285. − 652. i)18-s − 1.45e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.401 + 0.916i)2-s − 0.694·3-s + (−0.678 − 0.734i)4-s + (0.278 − 0.636i)6-s + 1.26i·7-s + (0.945 − 0.326i)8-s − 0.517·9-s − 0.801i·11-s + (0.470 + 0.510i)12-s − 0.210·13-s + (−1.15 − 0.506i)14-s + (−0.0802 + 0.996i)16-s − 1.77i·17-s + (0.207 − 0.474i)18-s − 0.924i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8067456775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8067456775\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.26 - 5.18i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.8T + 243T^{2} \) |
| 7 | \( 1 - 163. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 321. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 128.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.11e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.45e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.23e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.07e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.32e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.52e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.91e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.34e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.68e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e5iT - 8.58e9T^{2} \) |
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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
−11.57727030964230707547241802427, −10.97537728389862198093583553352, −9.380743637864804729194196062980, −8.944481095602208211421660594096, −7.70751310081882366346685616108, −6.50151149865202718424539430157, −5.58553545826570978203412032057, −4.94206316942828333971510718777, −2.77528312337700160078318744150, −0.68219193186695222540442461053,
0.53450501871742175373383690551, 1.90508697466420051823660322468, 3.66644819728538982814396477458, 4.58976484149877174709647541641, 6.10318984144880880150633877130, 7.47459620192632034870479045618, 8.378513843584146694571592360882, 9.784633939222596766878248370299, 10.50791458345212671437758373614, 11.14946475480647876059674584836
