L(s) = 1 | + (0.866 + 0.5i)2-s + (2.26 − 1.30i)3-s + (0.499 + 0.866i)4-s + (2.26 + 1.30i)5-s + 2.61·6-s + 0.999i·8-s + (1.91 − 3.31i)9-s + (1.30 + 2.26i)10-s + (1.89 − 2.72i)11-s + (2.26 + 1.30i)12-s − 5.54·13-s + 6.82·15-s + (−0.5 + 0.866i)16-s + (3.31 − 1.91i)18-s + (1.14 − 1.98i)19-s + 2.61i·20-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (1.30 − 0.754i)3-s + (0.249 + 0.433i)4-s + (1.01 + 0.584i)5-s + 1.06·6-s + 0.353i·8-s + (0.638 − 1.10i)9-s + (0.413 + 0.715i)10-s + (0.570 − 0.821i)11-s + (0.653 + 0.377i)12-s − 1.53·13-s + 1.76·15-s + (−0.125 + 0.216i)16-s + (0.781 − 0.451i)18-s + (0.263 − 0.456i)19-s + 0.584i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.030612193\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.030612193\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.89 + 2.72i)T \) |
good | 3 | \( 1 + (-2.26 + 1.30i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.26 - 1.30i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 + (-6.12 + 3.53i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + (5.73 + 3.31i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.24 - 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.5 - 6.08i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.14 + 1.98i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.171 + 0.297i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-3.24 - 5.62i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.951T + 83T^{2} \) |
| 89 | \( 1 + (10.6 + 6.15i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.09iT - 97T^{2} \) |
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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
−9.735233668728047719545410399381, −8.874595148698364924913070773970, −8.233411137330219100352609227817, −7.04499421100849798849046027145, −6.87661141585676355790844278506, −5.75077816110247855493894093457, −4.70956598500759253629908134838, −3.21528855666531913453485006106, −2.74487190171455090372457404747, −1.68956583434350997964898956204,
1.77154475825143230950704255317, 2.51479368893911919387212761862, 3.60601627167667709892177165548, 4.61443502239986268268393278714, 5.15448846545336109817887733469, 6.37806478926809353886705385056, 7.47188696736469543525623651506, 8.423002260229535295846435982257, 9.427628732194678863226984503821, 9.804195241958448523582686309933