L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s − 1.00i·4-s + (−0.258 + 0.965i)5-s + (0.258 + 0.965i)6-s + (−1.22 − 1.22i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.866 + 0.500i)12-s + (0.707 − 0.707i)13-s − 1.73·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1.36 − 1.36i)17-s + (−0.965 − 0.258i)18-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s − 1.00i·4-s + (−0.258 + 0.965i)5-s + (0.258 + 0.965i)6-s + (−1.22 − 1.22i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.866 + 0.500i)12-s + (0.707 − 0.707i)13-s − 1.73·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1.36 − 1.36i)17-s + (−0.965 − 0.258i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9452363476\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9452363476\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 47 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.814850345516566645352689907122, −9.096017393764000589123701308390, −7.44666370852734065482839517704, −6.83927408686370419818644468014, −5.91723307944549878465354523991, −5.26437283049197508100930530735, −3.88176596921532049422334476752, −3.60006914549651792146201059434, −2.84717386715660152968024220475, −0.63611146449134114639580549659,
1.73310815068788685387437250656, 3.10273314721915476193497053929, 4.04894625129553888989873840331, 5.31717009219395415875327398768, 5.84427966691479019290462172013, 6.34062973198834430759545862384, 7.33372601345514370876202692081, 8.204202673094879301240036280317, 8.776510363841565971432602227119, 9.511396837612868670601114868504