L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 2·5-s + (−0.499 − 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)10-s + 0.999·12-s + 1.99·14-s + (−1 + 1.73i)15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + 0.999·18-s + (3 + 5.19i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (−0.204 − 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + 0.288·12-s + 0.534·14-s + (−0.258 + 0.447i)15-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + 0.235·18-s + (0.688 + 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.317393546\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.317393546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.943605321589757556460227478288, −9.443730558809914709056438519300, −8.360181206716990615801872853068, −7.52603363573901195948023823579, −6.39493597316988838302423067692, −6.01155636052248047055835378070, −4.90839654789549741958747741100, −4.02858383587444260502653930298, −2.59038320550320366500691043304, −0.891627287176597631210563176832,
1.11010681609343546837981652507, 2.32694789004965129489021697840, 3.14297483034777175511540972106, 4.73882444032061030566361779909, 5.62213911526220617674109292968, 6.51002941598703281298339055995, 7.31934016651896655492546429919, 8.524952657966552030407085928034, 9.099885430832980605017371436170, 9.916990580057499581559890464604