| L(s) = 1 | + (3.80 − 1.23i)2-s + (−8.70 − 11.9i)3-s + (12.9 − 9.40i)4-s + (45.9 + 31.8i)5-s + (−47.9 − 34.8i)6-s − 136. i·7-s + (37.6 − 51.7i)8-s + (7.29 − 22.4i)9-s + (214. + 64.6i)10-s + (−100. − 309. i)11-s + (−225. − 73.2i)12-s + (−943. − 306. i)13-s + (−169. − 521. i)14-s + (−17.4 − 827. i)15-s + (79.1 − 243. i)16-s + (−80.3 + 110. i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.558 − 0.768i)3-s + (0.404 − 0.293i)4-s + (0.821 + 0.570i)5-s + (−0.543 − 0.394i)6-s − 1.05i·7-s + (0.207 − 0.286i)8-s + (0.0300 − 0.0923i)9-s + (0.676 + 0.204i)10-s + (−0.250 − 0.771i)11-s + (−0.451 − 0.146i)12-s + (−1.54 − 0.502i)13-s + (−0.230 − 0.710i)14-s + (−0.0200 − 0.949i)15-s + (0.0772 − 0.237i)16-s + (−0.0674 + 0.0927i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.34939 - 1.66593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34939 - 1.66593i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.80 + 1.23i)T \) |
| 5 | \( 1 + (-45.9 - 31.8i)T \) |
| good | 3 | \( 1 + (8.70 + 11.9i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 136. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (100. + 309. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (943. + 306. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (80.3 - 110. i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.82e3 - 1.32e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-4.52e3 + 1.47e3i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-2.59e3 + 1.88e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.75e3 - 2.72e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (6.51e3 + 2.11e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.93e3 - 1.21e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.06e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (4.63e3 + 6.38e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.34e4 - 1.85e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.48e3 - 7.66e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-264. - 815. i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-2.07e4 + 2.86e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (8.39e3 - 6.10e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.95e4 + 1.28e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (4.05e4 - 2.94e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.48e4 + 8.92e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.73e4 - 1.15e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (6.36e4 + 8.76e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−13.97008853163175675449639450134, −13.14144205431006268245563288813, −12.07946484900350263772925789499, −10.81192959445951952372773570063, −9.855465934019228896147530311474, −7.44471524576093277353210376492, −6.51905558082175662867091863847, −5.18744042168248691266272736165, −3.02201192410609749489947035539, −1.00533100291700537018132839101,
2.39544795914352223431935045913, 4.98064520197187713125496334384, 5.21241505018231184547418447378, 7.08221319988903293162679448702, 9.110837076116769263865862810810, 10.05464033219301044405121767069, 11.62256833048127903706951633703, 12.54175766924404862533365485873, 13.72570227920473692075451023206, 15.07225272843110173130928571122
