L(s) = 1 | − 2-s + (0.209 − 1.71i)3-s + 4-s + i·5-s + (−0.209 + 1.71i)6-s + 0.264·7-s − 8-s + (−2.91 − 0.719i)9-s − i·10-s + 2i·11-s + (0.209 − 1.71i)12-s − 5.43i·13-s − 0.264·14-s + (1.71 + 0.209i)15-s + 16-s − 3.28i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.120 − 0.992i)3-s + 0.5·4-s + 0.447i·5-s + (−0.0854 + 0.701i)6-s + 0.0999·7-s − 0.353·8-s + (−0.970 − 0.239i)9-s − 0.316i·10-s + 0.603i·11-s + (0.0603 − 0.496i)12-s − 1.50i·13-s − 0.0707·14-s + (0.443 + 0.0540i)15-s + 0.250·16-s − 0.796i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.467224 - 0.746656i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.467224 - 0.746656i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.209 + 1.71i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-1.41 + 4.12i)T \) |
good | 7 | \( 1 - 0.264T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 5.43iT - 13T^{2} \) |
| 17 | \( 1 + 3.28iT - 17T^{2} \) |
| 23 | \( 1 + 3.96iT - 23T^{2} \) |
| 29 | \( 1 - 0.418T + 29T^{2} \) |
| 31 | \( 1 + 3.56iT - 31T^{2} \) |
| 37 | \( 1 + 0.307iT - 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 - 0.666iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 8.08iT - 79T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 1.47iT - 97T^{2} \) |
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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
−10.45675653243975267569808135310, −9.532027114289653926260719150143, −8.545683495572799147731698445947, −7.70452167699220949313481575430, −7.10739642860083457439317237496, −6.19612950072949939757267547128, −5.07156229072422456816310673147, −3.15576146990859265651399961220, −2.27382941578790942647362525781, −0.62111438818050233261484711067,
1.69806550904465297243744247141, 3.34354675687054925249476296885, 4.33417152981229313593279279001, 5.51302340690430417028363260918, 6.44966189047073980372680076053, 7.80919820650948512233436116547, 8.596076132904021843047446335959, 9.281208635566464104122007587449, 9.973118112397057509966944081811, 10.91823436509602004249527431176