L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s − 3.46i·7-s + 2.82·8-s + 3.16·11-s + 4.47·13-s + (−4.24 + 2.44i)14-s + (−2.00 − 3.46i)16-s − 6.32·17-s + (−2 + 3.87i)19-s + (−2.23 − 3.87i)22-s − 5.47i·23-s + 5·25-s + (−3.16 − 5.47i)26-s + (5.99 + 3.46i)28-s − 1.41·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.30i·7-s + 0.999·8-s + 0.953·11-s + 1.24·13-s + (−1.13 + 0.654i)14-s + (−0.500 − 0.866i)16-s − 1.53·17-s + (−0.458 + 0.888i)19-s + (−0.476 − 0.825i)22-s − 1.14i·23-s + 25-s + (−0.620 − 1.07i)26-s + (1.13 + 0.654i)28-s − 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.228235337\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.228235337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2 - 3.87i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 23 | \( 1 + 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.47iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 7.74iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−9.378725031958557990279982906803, −8.578727937259145338571168297554, −8.044917750795123303290660680387, −6.81730279935339114289356169239, −6.38540896212653828488979665420, −4.51129243309391534826406825109, −4.15967133074733944696950561325, −3.16082571277051090695649680488, −1.75852462265135540497199257052, −0.69985080228520100585745762067,
1.26942789262007981454662609880, 2.58655123966279304473635688526, 4.11614268779359623289803513671, 4.96610182901642757204941847891, 6.13248554467394205240035440387, 6.35241226579316721164558190025, 7.38093814930307960512034304945, 8.594232796022659616061441981131, 8.847754946175079366252091613531, 9.391677620507062123913877134801