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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.226953 + 1.35205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.226953 + 1.35205i\) |
\(L(\frac{3}{2})\) |
|
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
−10.38417841277629524222631421617, −9.422013955576008874731675919394, −8.714464574936282902891143256001, −8.069355063536038073311953202122, −7.29788315267811998727009419356, −6.28820165258194532029230110490, −5.39982188691544057403922058462, −4.78103976498932175872321903935, −2.86345210748665858378528249489, −1.91711729101855349514306535483,
0.851816672661626482220210886722, 1.63215417421461518845402725940, 3.15038477548417246592115623155, 4.14199261643933297261626175206, 5.29003017500586119776148896038, 6.52304717602297599711678249883, 7.23689921502422585531885153363, 8.302205944972082362720160285367, 9.326058466203797938420048307780, 9.885011139378536373738290564410