L(s) = 1 | + (0.747 + 1.29i)2-s + (−0.0473 − 0.0820i)3-s + (−0.118 + 0.204i)4-s − 5-s + (0.0708 − 0.122i)6-s + (−2.41 + 4.18i)7-s + 2.63·8-s + (1.49 − 2.59i)9-s + (−0.747 − 1.29i)10-s + (0.534 + 0.926i)11-s + 0.0224·12-s − 7.21·14-s + (0.0473 + 0.0820i)15-s + (2.20 + 3.82i)16-s + (−1.77 + 3.08i)17-s + 4.47·18-s + ⋯ |
L(s) = 1 | + (0.528 + 0.915i)2-s + (−0.0273 − 0.0473i)3-s + (−0.0591 + 0.102i)4-s − 0.447·5-s + (0.0289 − 0.0501i)6-s + (−0.912 + 1.57i)7-s + 0.932·8-s + (0.498 − 0.863i)9-s + (−0.236 − 0.409i)10-s + (0.161 + 0.279i)11-s + 0.00647·12-s − 1.92·14-s + (0.0122 + 0.0211i)15-s + (0.552 + 0.956i)16-s + (−0.431 + 0.747i)17-s + 1.05·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.708114 + 1.57847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.708114 + 1.57847i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.747 - 1.29i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (2.41 - 4.18i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.534 - 0.926i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.77 - 3.08i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.86 - 4.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.54 - 6.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + (-0.0126 - 0.0219i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.133 + 0.232i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.77 - 3.08i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 - 0.991T + 53T^{2} \) |
| 59 | \( 1 + (4.36 - 7.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.58 + 4.48i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.88 + 6.72i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 + 0.725T + 83T^{2} \) |
| 89 | \( 1 + (6.75 + 11.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.71 + 2.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.39283408855286840768475887671, −9.489135260403753079089726457099, −8.785733816743036635999145971431, −7.75017867606937020077034453451, −6.80317567621797898968881347230, −6.12703767372872699904383523927, −5.56012386456583112452964924578, −4.30431231964684314842256222343, −3.36297355758466338405132236801, −1.80008660751092900441386295726,
0.72360094723884679659319628720, 2.41110338092825419507978399886, 3.43816638033658784201803179639, 4.30437528437193980749923665895, 4.87652630389837023098961067091, 6.77860838695208586113891025086, 7.10365137130797223270291180692, 8.072286575832415679876346413152, 9.287893691814625267466331615660, 10.31885045804464693497015154145