| L(s) = 1 | + i·2-s − 4-s − 2.70·5-s + (−2.02 − 1.70i)7-s − i·8-s − 2.70i·10-s + 4.07i·11-s − 1.53i·13-s + (1.70 − 2.02i)14-s + 16-s − 3.59·17-s − 1.31i·19-s + 2.70·20-s − 4.07·22-s − 4.52i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.20·5-s + (−0.763 − 0.645i)7-s − 0.353i·8-s − 0.854i·10-s + 1.22i·11-s − 0.427i·13-s + (0.456 − 0.539i)14-s + 0.250·16-s − 0.872·17-s − 0.302i·19-s + 0.604·20-s − 0.868·22-s − 0.942i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0864 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6522081614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6522081614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.02 + 1.70i)T \) |
| 29 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 4.07iT - 11T^{2} \) |
| 13 | \( 1 + 1.53iT - 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 1.31iT - 19T^{2} \) |
| 23 | \( 1 + 4.52iT - 23T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 1.93T + 37T^{2} \) |
| 41 | \( 1 - 1.48T + 41T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 - 0.571T + 47T^{2} \) |
| 53 | \( 1 - 1.02iT - 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 1.36iT - 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 - 9.26iT - 71T^{2} \) |
| 73 | \( 1 - 5.35iT - 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.631T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 4.74iT - 97T^{2} \) |
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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
−8.515623338283448386415137993890, −7.79877932578949379609994511067, −7.22600580156841471388986381719, −6.73311228015408641465639301893, −5.88807600369328045403319944556, −4.66787497137447347622285787320, −4.28586260066536500349631362299, −3.51327697087295210426728454624, −2.36140659581341688306519918397, −0.60289978332135820497868237214,
0.34790949840451098875352506980, 1.78222621127581948379186654696, 3.08927209289557819866835924329, 3.42876619190110680144772923156, 4.30798418995560472672677435960, 5.23867332568217091233496192168, 6.10967746504062140717410966283, 6.88763191215141102142654410496, 7.79395584788327945078594710597, 8.613386851124346091341016992062
