L(s) = 1 | − 4.85i·2-s − 5.19·3-s − 7.60·4-s − 12.5·5-s + 25.2i·6-s + 50.4·7-s − 40.7i·8-s + 27·9-s + 60.7i·10-s − 191. i·11-s + 39.5·12-s + 0.640i·13-s − 245. i·14-s + 64.9·15-s − 319.·16-s − 549.·17-s + ⋯ |
L(s) = 1 | − 1.21i·2-s − 0.577·3-s − 0.475·4-s − 0.500·5-s + 0.701i·6-s + 1.03·7-s − 0.637i·8-s + 0.333·9-s + 0.607i·10-s − 1.58i·11-s + 0.274·12-s + 0.00378i·13-s − 1.25i·14-s + 0.288·15-s − 1.24·16-s − 1.90·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8827584366\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8827584366\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (2.56e3 + 2.35e3i)T \) |
good | 2 | \( 1 + 4.85iT - 16T^{2} \) |
| 5 | \( 1 + 12.5T + 625T^{2} \) |
| 7 | \( 1 - 50.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 191. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 0.640iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 549.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 524.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 543. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 296.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.07e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.16e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.12e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.52e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 103. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.81e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.90e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.85e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.82e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.75e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 9.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.73e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.52e4iT - 8.85e7T^{2} \) |
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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
−11.42657774274364147221727730923, −10.96620474367873240343979069944, −9.666221475634662861681406974239, −8.489157963379956669022709543432, −7.27462194803255119237310434384, −5.84117925526934022373101696806, −4.47632696705323987792357381022, −3.30354995418928085646263146949, −1.68374632806477959159921473877, −0.34408059119288353359247913256,
1.96276178361754431393989884469, 4.49153071848316222834369384884, 5.09069105936178914753412651429, 6.55240309825810778810872814013, 7.32561858001620447958392100467, 8.151320607778477501484488863619, 9.384404728907593000729619105551, 10.86783180418615424347161559498, 11.57206485892255804475809115202, 12.54087975856227084353416752915