L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.866 + 0.499i)20-s − 25-s + (−0.366 + 0.366i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 + 0.5i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.866 + 0.499i)20-s − 25-s + (−0.366 + 0.366i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 + 0.5i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6906038458\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6906038458\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 - 1.73T + T^{2} \) |
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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
−10.52364472738008781864003336432, −9.865580808505125747071357205473, −9.141986826820556767599086703799, −7.955177587201249068206129938279, −7.36354870971581762264830516389, −6.46479103134850073221741867070, −5.84606518034813986154252965710, −4.35728177321846310550292871526, −3.03133711333930172747107426393, −1.59090044178183410278437293239,
1.15443793449386252972175086115, 2.42304464500207117343682442367, 3.91098326614564103591735517081, 4.81917691633289795937113065316, 6.07679696028216140005878011853, 7.48802752535027932945038149750, 7.76335069770711380815944705751, 8.997046376974766708220561228387, 9.460046933865326854344466571190, 10.31285639807901548577887021552