L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s + (2 + 1.73i)7-s − 0.999·8-s + (0.999 − 1.73i)10-s + (0.5 − 0.866i)11-s + 2·13-s + (−0.499 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (1.5 + 2.59i)19-s + 1.99·20-s + 0.999·22-s + (−0.5 − 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (0.755 + 0.654i)7-s − 0.353·8-s + (0.316 − 0.547i)10-s + (0.150 − 0.261i)11-s + 0.554·13-s + (−0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + (0.344 + 0.596i)19-s + 0.447·20-s + 0.213·22-s + (−0.104 − 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.045922423\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.045922423\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.353339777482889571855812742081, −8.634241229400959048779810922337, −8.132893272559658981776450330375, −7.38270641670856959585059276384, −6.17767691890968515030889519735, −5.55614596119182970154328984323, −4.64869217820727699757302239422, −3.95713935015641849007491838743, −2.64825639342696373010474300652, −1.08782965576418713307629610928,
1.00861060845835391989695822450, 2.32273083633484004547738009776, 3.45693104476761686853482530834, 4.16200257511938105059685924192, 5.08098557385388890963467800026, 6.14187512450888232658619468630, 7.13245301431707846490589177004, 7.72491313699978643677608821024, 8.748251275942159686248751587276, 9.621520491354685308370318909306