| L(s) = 1 | + (−1.04 + 0.954i)2-s + (0.179 − 1.99i)4-s − i·5-s − 0.851i·7-s + (1.71 + 2.25i)8-s + (0.954 + 1.04i)10-s − 1.56·11-s + 4.53·13-s + (0.812 + 0.889i)14-s + (−3.93 − 0.713i)16-s − 4.69i·17-s − 7.37i·19-s + (−1.99 − 0.179i)20-s + (1.63 − 1.49i)22-s − 5.72·23-s + ⋯ |
| L(s) = 1 | + (−0.738 + 0.674i)2-s + (0.0895 − 0.995i)4-s − 0.447i·5-s − 0.321i·7-s + (0.605 + 0.795i)8-s + (0.301 + 0.330i)10-s − 0.473·11-s + 1.25·13-s + (0.217 + 0.237i)14-s + (−0.983 − 0.178i)16-s − 1.13i·17-s − 1.69i·19-s + (−0.445 − 0.0400i)20-s + (0.349 − 0.319i)22-s − 1.19·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0895 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0895 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7436422340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7436422340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 - 0.954i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 0.851iT - 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 4.69iT - 17T^{2} \) |
| 19 | \( 1 + 7.37iT - 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 7.57iT - 29T^{2} \) |
| 31 | \( 1 - 2.98iT - 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 - 3.99iT - 41T^{2} \) |
| 43 | \( 1 + 6.88iT - 43T^{2} \) |
| 47 | \( 1 - 0.279T + 47T^{2} \) |
| 53 | \( 1 + 0.417iT - 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 + 6.24iT - 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.63iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 8.75iT - 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.861775017369337912617908912265, −8.648091901634731030084708915502, −7.51435856884280084895915359723, −6.98492489972165669481010726911, −6.03524594458295762928511635547, −5.18598576850259293678945545435, −4.44083527806970941079961760812, −3.01535650768270860180517748735, −1.60278216936275793718256451622, −0.37764515045267162225840602027,
1.47696767214529912271625334791, 2.40829847223698735211097357946, 3.62475044641080305602934634711, 4.12269332820396369643036719745, 5.85686722251017453400121405867, 6.26193290498879307328650867785, 7.62268740776088616011547937116, 8.107998321638882203163222860476, 8.749877354902033427484775313627, 9.808250948509443825098932637892
