L(s) = 1 | + (1.14 + 1.14i)2-s + (−0.891 − 0.453i)3-s + 1.61i·4-s + (−0.5 − 1.53i)6-s + (0.831 − 0.831i)7-s + (−0.707 + 0.707i)8-s + (0.587 + 0.809i)9-s + (0.734 − 1.44i)12-s + 1.90·14-s + (−0.437 − 0.437i)17-s + (−0.253 + 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (−0.156 − 0.987i)27-s + (1.34 + 1.34i)28-s + ⋯ |
L(s) = 1 | + (1.14 + 1.14i)2-s + (−0.891 − 0.453i)3-s + 1.61i·4-s + (−0.5 − 1.53i)6-s + (0.831 − 0.831i)7-s + (−0.707 + 0.707i)8-s + (0.587 + 0.809i)9-s + (0.734 − 1.44i)12-s + 1.90·14-s + (−0.437 − 0.437i)17-s + (−0.253 + 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (−0.156 − 0.987i)27-s + (1.34 + 1.34i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.994146304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.994146304\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.891 + 0.453i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.14 - 1.14i)T + iT^{2} \) |
| 7 | \( 1 + (-0.831 + 0.831i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.437 + 0.437i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.34 + 1.34i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1.14 + 1.14i)T - iT^{2} \) |
| 59 | \( 1 + 1.17T + T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.90iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 0.618iT - T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 + 1.17T + T^{2} \) |
| 97 | \( 1 + (0.831 - 0.831i)T - iT^{2} \) |
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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
−8.315134255267933022568794722386, −7.75951567093428211191023820581, −7.08489424180831903691396562391, −6.69459170445996222052084749473, −5.69097070680443832627514337718, −5.30315569301860987681235758710, −4.33527532452211928671698805646, −4.06608416210623758947861418484, −2.51863708552352497970132548478, −1.11550402676002248517767201925,
1.27530065763881183560254966616, 2.23697246251203749123268717835, 3.18537833496421731620429209274, 4.24773921251716701762991926081, 4.61109482687593992136798640666, 5.50289041865698708816646304301, 5.87429826073502142270125284948, 6.82786611353287901644162565281, 7.994045245457298356653322214981, 8.840560457636217420773971301291