L(s) = 1 | + (2.31 + 0.619i)2-s + (−1.28 + 1.16i)3-s + (3.23 + 1.86i)4-s + (−1.69 + 1.69i)5-s + (−3.68 + 1.89i)6-s + (0.366 + 1.36i)7-s + (2.93 + 2.93i)8-s + (0.292 − 2.98i)9-s + (−4.96 + 2.86i)10-s + (−0.453 + 1.69i)11-s + (−6.31 + 1.36i)12-s + 3.38i·14-s + (0.202 − 4.14i)15-s + (1.23 + 2.13i)16-s + (−1.07 + 1.85i)17-s + (2.52 − 6.72i)18-s + ⋯ |
L(s) = 1 | + (1.63 + 0.438i)2-s + (−0.740 + 0.671i)3-s + (1.61 + 0.933i)4-s + (−0.757 + 0.757i)5-s + (−1.50 + 0.773i)6-s + (0.138 + 0.516i)7-s + (1.03 + 1.03i)8-s + (0.0975 − 0.995i)9-s + (−1.56 + 0.906i)10-s + (−0.136 + 0.510i)11-s + (−1.82 + 0.394i)12-s + 0.904i·14-s + (0.0523 − 1.06i)15-s + (0.308 + 0.533i)16-s + (−0.260 + 0.450i)17-s + (0.595 − 1.58i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.989649 + 2.15771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.989649 + 2.15771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 - 1.16i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-2.31 - 0.619i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (1.69 - 1.69i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.366 - 1.36i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.453 - 1.69i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.07 - 1.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 0.267i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.79 + 2.76i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.46 - 4.46i)T + 31iT^{2} \) |
| 37 | \( 1 + (-6.59 - 1.76i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.619 + 0.166i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (7.09 + 4.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.77 - 6.77i)T + 47iT^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 + (-4.62 + 1.23i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.26 + 8.46i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.23 - 4.62i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (6.09 - 6.09i)T - 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.60 - 9.70i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (12.5 - 3.36i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.53254100878788825112551794939, −10.75095670812111744631991135007, −9.723797071422578561166484555556, −8.271240510306957515078025796766, −7.06778832021740928805006146707, −6.44127024277197400660879075939, −5.46327029448833580841306054887, −4.60782039115767170561478164324, −3.78972020144447097862363621864, −2.76055620826158948035515379256,
0.965573296617319730461886603877, 2.63880398825203433008209367978, 4.07944284715895810262877600491, 4.76093590683125014171186865945, 5.65082172953371529052664087690, 6.59701927869960176501888166406, 7.57839042075660000472324403434, 8.559479416596111546134271371649, 10.25146170513644856508520284449, 11.11514874673225056590172853758