L(s) = 1 | − i·2-s + 4-s + (−2.12 − 0.707i)5-s − 1.41·7-s − 3i·8-s + (−0.707 + 2.12i)10-s + (−3 − 1.41i)11-s − 2.82·13-s + 1.41i·14-s − 16-s − 2i·17-s − 4.24i·19-s + (−2.12 − 0.707i)20-s + (−1.41 + 3i)22-s + (3.99 + 3i)25-s + 2.82i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (−0.948 − 0.316i)5-s − 0.534·7-s − 1.06i·8-s + (−0.223 + 0.670i)10-s + (−0.904 − 0.426i)11-s − 0.784·13-s + 0.377i·14-s − 0.250·16-s − 0.485i·17-s − 0.973i·19-s + (−0.474 − 0.158i)20-s + (−0.301 + 0.639i)22-s + (0.799 + 0.600i)25-s + 0.554i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0799393 - 0.825575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0799393 - 0.825575i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.12 + 0.707i)T \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.7iT - 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.83962649329707433387820493049, −9.754577071933801123923556980020, −8.925037978472295233649812206285, −7.58688429935044212194813636487, −7.16084624731319949304994796171, −5.78693712587176488909752723029, −4.54318828558047392610734227358, −3.36035462887173619794007846140, −2.46160326819386650833251257065, −0.45896238528211324173834952690,
2.32613313963983107035790728678, 3.52445723541452860436046070428, 4.88303839130992806717002862225, 5.96968731911578371593414476857, 6.93169502524740171865893712882, 7.68883500419090755738146469052, 8.215214185695769612336334378516, 9.628180454876689618824586119943, 10.56240817795265914917765285952, 11.29402191827050929810804189111