L(s) = 1 | − i·2-s + (1.22 − 1.22i)3-s − 4-s + (−1.22 − 1.22i)6-s + (1 + 2.44i)7-s + i·8-s − 2.99i·9-s + (−1.22 + 1.22i)12-s − 2.44i·13-s + (2.44 − i)14-s + 16-s + 4.89·17-s − 2.99·18-s − 2.44i·19-s + (4.22 + 1.77i)21-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.707 − 0.707i)3-s − 0.5·4-s + (−0.499 − 0.499i)6-s + (0.377 + 0.925i)7-s + 0.353i·8-s − 0.999i·9-s + (−0.353 + 0.353i)12-s − 0.679i·13-s + (0.654 − 0.267i)14-s + 0.250·16-s + 1.18·17-s − 0.707·18-s − 0.561i·19-s + (0.921 + 0.387i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.002362311\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.002362311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.635827714905113691992084671367, −8.733149434662089782934572661821, −8.218453814674227908848543914294, −7.39486794251593951140750361101, −6.18297487871858723343222126833, −5.34889516316225397284979613699, −4.11181662973054219296095248027, −2.91728346294052171119198843118, −2.30483622545207201596114306436, −0.925177338972005694605538232554,
1.53570434389176955491284794631, 3.28357244760207935136789229141, 4.02733402092110112665028601119, 4.92519658321201342349905784608, 5.80037836963358144916291297199, 7.13212304164963986819618312753, 7.65823471511444656801797511349, 8.431497675000231453367696050767, 9.342008312214282539744113738176, 9.967387146907974064381768134360