L(s) = 1 | + (0.707 + 0.707i)3-s + 1.41i·7-s + 1.00i·9-s + (−1 − i)13-s + 6·17-s + (4.24 + 4.24i)19-s + (−1.00 + 1.00i)21-s + 8.48i·23-s + 5i·25-s + (−0.707 + 0.707i)27-s + (−6 − 6i)29-s + 1.41·31-s + (−5 + 5i)37-s − 1.41i·39-s − 6i·41-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + 0.534i·7-s + 0.333i·9-s + (−0.277 − 0.277i)13-s + 1.45·17-s + (0.973 + 0.973i)19-s + (−0.218 + 0.218i)21-s + 1.76i·23-s + i·25-s + (−0.136 + 0.136i)27-s + (−1.11 − 1.11i)29-s + 0.254·31-s + (−0.821 + 0.821i)37-s − 0.226i·39-s − 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43869 + 0.961306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43869 + 0.961306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (-4.24 - 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (6 + 6i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + (-6 + 6i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (5 + 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−10.12772495389654508473594890036, −9.759425247332350924728365390144, −8.880423423465933409048114896211, −7.82631755983442344777800940703, −7.34273556707219209599135898255, −5.64045563084160013897518394465, −5.41328411626768394114407898875, −3.81646332914554207606022922929, −3.10699558274464834045359739424, −1.62194819701675941074547461307,
0.913798529720631648626278487883, 2.45936526432150365554978538210, 3.53798438272016768714008025277, 4.67847701625107412033620927682, 5.76024838573185380702124064153, 6.94942144351404873561573091195, 7.43931293830525321429792601789, 8.453797947080318434274886697259, 9.240473299001224430180973587217, 10.18332188222451376883475016964