| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.386 − 2.19i)5-s + (−0.939 − 0.342i)6-s + (0.326 + 0.565i)7-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−1.70 + 1.43i)10-s + (0.766 − 1.32i)11-s + (0.499 + 0.866i)12-s + (0.439 + 0.160i)13-s + (0.113 − 0.642i)14-s + (−0.386 − 2.19i)15-s + (−0.939 + 0.342i)16-s + (−1.61 − 1.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.172 − 0.980i)5-s + (−0.383 − 0.139i)6-s + (0.123 + 0.213i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.539 + 0.452i)10-s + (0.230 − 0.400i)11-s + (0.144 + 0.250i)12-s + (0.121 + 0.0443i)13-s + (0.0302 − 0.171i)14-s + (−0.0998 − 0.566i)15-s + (−0.234 + 0.0855i)16-s + (−0.391 − 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.847582 - 0.488983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847582 - 0.488983i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (2.23 - 3.74i)T \) |
| good | 5 | \( 1 + (-0.386 + 2.19i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.766 + 1.32i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.439 - 0.160i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.61 + 1.35i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.02 - 5.81i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.38 - 5.35i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.31 - 7.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + (3.26 - 1.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.78 + 10.1i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.55 - 2.98i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.07 + 11.7i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 9.14i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.58 + 8.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.190 + 0.160i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.772 - 4.38i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-8.54 + 3.10i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (11.3 - 4.12i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.85 + 3.21i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (15.7 + 5.73i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.89 - 1.58i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.22673979144617759829149129863, −12.42371971246208890371917827382, −11.39776259370537352342896271806, −10.07721204178323785431801535419, −8.926775194570996470817810122272, −8.453420369961900975038998033556, −7.02887408707105239057327372450, −5.27375097732845350294000050430, −3.56868407894679130114210868349, −1.64200822936618385186842858496,
2.47989876183043574185938126294, 4.36028722621144727898319685301, 6.25259008286742400940574054806, 7.19677649520715541997633418116, 8.354825801656017323594847456615, 9.487682231286897692625489882255, 10.46278388897708711792008334841, 11.30782229304229200177102418214, 12.98930770664270173886092574096, 14.04808887801591542324945659769
