| L(s) = 1 | + (−0.325 − 0.0373i)2-s + (1.10 + 0.803i)3-s + (−1.84 − 0.428i)4-s + (−0.564 + 0.825i)5-s + (−0.329 − 0.302i)6-s + (0.602 + 0.619i)7-s + (1.20 + 0.428i)8-s + (−0.349 − 1.07i)9-s + (0.214 − 0.247i)10-s + (−3.25 − 0.610i)11-s + (−1.69 − 1.95i)12-s + (−2.55 + 4.23i)13-s + (−0.172 − 0.224i)14-s + (−1.28 + 0.459i)15-s + (3.02 + 1.48i)16-s + (−0.489 + 0.400i)17-s + ⋯ |
| L(s) = 1 | + (−0.230 − 0.0263i)2-s + (0.638 + 0.464i)3-s + (−0.921 − 0.214i)4-s + (−0.252 + 0.369i)5-s + (−0.134 − 0.123i)6-s + (0.227 + 0.234i)7-s + (0.424 + 0.151i)8-s + (−0.116 − 0.358i)9-s + (0.0678 − 0.0782i)10-s + (−0.982 − 0.184i)11-s + (−0.489 − 0.564i)12-s + (−0.707 + 1.17i)13-s + (−0.0461 − 0.0598i)14-s + (−0.332 + 0.118i)15-s + (0.755 + 0.371i)16-s + (−0.118 + 0.0970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0471847 + 0.390630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0471847 + 0.390630i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (3.25 + 0.610i)T \) |
| good | 2 | \( 1 + (0.325 + 0.0373i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-1.10 - 0.803i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-0.602 - 0.619i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (2.55 - 4.23i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (0.489 - 0.400i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (4.94 + 0.282i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (1.00 - 6.97i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-1.77 + 4.55i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (8.67 + 3.66i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.534 - 6.21i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (5.07 - 2.86i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (3.38 - 0.994i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (0.667 - 3.29i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (9.32 - 4.58i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-5.27 - 2.97i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (-6.49 + 0.745i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (-5.93 + 12.9i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.575 - 2.18i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-5.66 - 10.7i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.475 - 16.6i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (-15.4 - 2.67i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (0.710 + 0.456i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.31 + 1.92i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.92027728840681937967201766491, −9.809791227963695194619667440709, −9.482272078101939612112720338042, −8.459934224131787432012461034755, −7.909013450435827645357502432700, −6.62796272164117703473322711978, −5.38753519858964357149142932093, −4.39904059446755490232941751235, −3.53392833573391043762920633632, −2.14083051898481348794666270746,
0.21078184525772268433999806109, 2.15254277916585124636129229158, 3.41669742344159672784112056724, 4.75571940993842283593829986133, 5.31592883518445630474013846268, 7.06383712021645969116470096416, 7.908212625320680934236223907891, 8.350469793168251973883213824654, 9.086398025199398124782012746325, 10.38445334269797838863609510834
