| L(s) = 1 | + (0.00789 − 0.0902i)2-s + (3.93 + 0.693i)4-s + (−0.474 + 4.97i)5-s + (7.83 + 5.48i)7-s + (0.187 − 0.699i)8-s + (0.445 + 0.0821i)10-s + (−2.28 − 0.832i)11-s + (0.641 + 7.33i)13-s + (0.557 − 0.663i)14-s + (14.9 + 5.43i)16-s + (−3.64 − 13.6i)17-s + (−15.5 + 8.98i)19-s + (−5.31 + 19.2i)20-s + (−0.0932 + 0.199i)22-s + (−14.0 + 9.81i)23-s + ⋯ |
| L(s) = 1 | + (0.00394 − 0.0451i)2-s + (0.982 + 0.173i)4-s + (−0.0949 + 0.995i)5-s + (1.11 + 0.783i)7-s + (0.0234 − 0.0874i)8-s + (0.0445 + 0.00821i)10-s + (−0.207 − 0.0756i)11-s + (0.0493 + 0.564i)13-s + (0.0397 − 0.0474i)14-s + (0.933 + 0.339i)16-s + (−0.214 − 0.801i)17-s + (−0.819 + 0.473i)19-s + (−0.265 + 0.961i)20-s + (−0.00423 + 0.00908i)22-s + (−0.609 + 0.426i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.83279 + 1.31129i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83279 + 1.31129i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.474 - 4.97i)T \) |
| good | 2 | \( 1 + (-0.00789 + 0.0902i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-7.83 - 5.48i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (2.28 + 0.832i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-0.641 - 7.33i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (3.64 + 13.6i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (15.5 - 8.98i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (14.0 - 9.81i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-10.6 - 12.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-7.88 + 44.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (3.07 + 11.4i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-36.2 - 30.4i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-7.28 - 15.6i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-56.3 - 39.4i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-21.4 - 21.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (31.6 + 86.8i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (10.2 + 58.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-101. + 8.90i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (42.6 - 73.8i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (25.2 - 94.4i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (97.2 + 115. i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-153. - 13.4i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (46.0 - 26.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-22.5 + 10.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.34995887690314969443579192074, −10.61288371131156501706363008075, −9.491561750277963463406167717127, −8.185676817346177110732372641396, −7.55479090619630450728741552937, −6.50440091631190885313442051735, −5.67949903946561637428290333344, −4.19635305871253658780898295783, −2.73418107792888920583609387084, −1.93757446926649697397552934913,
1.00636853051191015992346117610, 2.20138717942535265604024318955, 4.00844002814637545637567982841, 4.99584645255810337705617029063, 6.04677848955455802957649568059, 7.24686561333589430294325596638, 8.068769665572986861370775224513, 8.774526624787911282166642644502, 10.40983278166668457309284848426, 10.65195155189758153232658618187
