L(s) = 1 | + 2.43·2-s + (−0.376 − 0.652i)3-s + 3.91·4-s + (−0.170 − 0.295i)5-s + (−0.916 − 1.58i)6-s + 4.65·8-s + (1.21 − 2.10i)9-s + (−0.415 − 0.719i)10-s + (1.21 + 2.10i)11-s + (−1.47 − 2.55i)12-s + (2.50 + 2.59i)13-s + (−0.128 + 0.222i)15-s + 3.49·16-s − 1.94·17-s + (2.95 − 5.12i)18-s + (3.14 − 5.44i)19-s + ⋯ |
L(s) = 1 | + 1.71·2-s + (−0.217 − 0.376i)3-s + 1.95·4-s + (−0.0763 − 0.132i)5-s + (−0.374 − 0.647i)6-s + 1.64·8-s + (0.405 − 0.702i)9-s + (−0.131 − 0.227i)10-s + (0.366 + 0.635i)11-s + (−0.425 − 0.737i)12-s + (0.693 + 0.720i)13-s + (−0.0332 + 0.0575i)15-s + 0.874·16-s − 0.472·17-s + (0.697 − 1.20i)18-s + (0.721 − 1.24i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.64728 - 0.884989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.64728 - 0.884989i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.50 - 2.59i)T \) |
good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + (0.376 + 0.652i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.170 + 0.295i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 + (2.22 - 3.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.987 + 1.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 + (6.26 - 10.8i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.20 - 7.28i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.50 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.746 - 1.29i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 0.626T + 59T^{2} \) |
| 61 | \( 1 + (-0.571 + 0.990i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.74 + 8.22i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.95 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.23 + 3.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (5.13 + 8.90i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.05038810128642534973445110496, −9.737637830554316423351957163948, −8.780743177160546556945942434453, −7.27963147305081769627625278650, −6.68002952262707555172482056629, −6.00517874651526705140915202405, −4.75756922848531031708232170027, −4.15392921758725349620203313955, −3.02203157493796540589639431562, −1.60351469940836083128015690356,
1.95099558375382084106402016846, 3.48674802619464543323935952763, 3.93284986339965082057327462709, 5.29531250006584164448681192751, 5.64776514438598218004393407531, 6.76823488227840940899961587427, 7.70838196413690120060974648499, 8.850394954106777423045605869432, 10.33761155564586333907837675783, 10.74833892856182825649139863565