| L(s) = 1 | + (1.40 − 0.150i)2-s + (0.961 + 1.16i)3-s + (1.95 − 0.424i)4-s + (0.926 − 2.03i)5-s + (1.52 + 1.48i)6-s + (−1.57 + 0.511i)7-s + (2.68 − 0.892i)8-s + (0.136 − 0.712i)9-s + (0.995 − 3.00i)10-s + (−0.244 + 1.93i)11-s + (2.37 + 1.86i)12-s + (0.678 − 3.55i)13-s + (−2.13 + 0.957i)14-s + (3.25 − 0.880i)15-s + (3.63 − 1.65i)16-s + (0.998 + 3.88i)17-s + ⋯ |
| L(s) = 1 | + (0.994 − 0.106i)2-s + (0.554 + 0.670i)3-s + (0.977 − 0.212i)4-s + (0.414 − 0.910i)5-s + (0.623 + 0.607i)6-s + (−0.595 + 0.193i)7-s + (0.948 − 0.315i)8-s + (0.0453 − 0.237i)9-s + (0.314 − 0.949i)10-s + (−0.0736 + 0.582i)11-s + (0.684 + 0.537i)12-s + (0.188 − 0.985i)13-s + (−0.571 + 0.255i)14-s + (0.840 − 0.227i)15-s + (0.909 − 0.414i)16-s + (0.242 + 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.77675 - 0.319435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.77675 - 0.319435i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.150i)T \) |
| 5 | \( 1 + (-0.926 + 2.03i)T \) |
| good | 3 | \( 1 + (-0.961 - 1.16i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (1.57 - 0.511i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.244 - 1.93i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.678 + 3.55i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.998 - 3.88i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (-3.62 - 2.99i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (0.187 - 0.199i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (8.04 - 0.506i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (-3.64 + 0.935i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-0.549 - 0.302i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (-4.18 + 3.93i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (2.86 - 2.08i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (1.07 + 2.70i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (4.21 - 6.64i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (7.93 - 3.73i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (2.62 - 2.79i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.486 - 7.73i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (-0.739 + 0.292i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (6.33 + 2.98i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (5.38 + 6.51i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (10.7 - 12.9i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-1.69 + 3.59i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (12.7 - 0.803i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.923132419454980076908597453010, −9.377945510572531645879774096984, −8.310829854130163629933526587744, −7.43872534398711220294519392282, −6.09637999974002785227371626661, −5.62280710237232928143968561822, −4.53590231097152987706934857430, −3.72935839912698036099601906424, −2.87545771779872741760030204230, −1.45650113851618403285077900963,
1.74330715539233941769727910914, 2.81081242973885897246314650880, 3.39791357740460760898669376426, 4.75587346374514076386769134495, 5.83202331865884064921479055496, 6.69927848669398808100480508764, 7.20641406928336585097892840151, 7.948775163217514099029881124950, 9.231964246335183628990163670575, 10.04148458055160790107513848029
