| L(s) = 1 | + (1.22 − 1.22i)3-s − 2.44i·7-s − 2.99i·9-s − 4.89·11-s + 2·13-s − 6i·17-s + 4.89i·19-s + (−2.99 − 2.99i)21-s − 2.44·23-s + (−3.67 − 3.67i)27-s − 9.79i·31-s + (−5.99 + 5.99i)33-s − 2·37-s + (2.44 − 2.44i)39-s + 6i·41-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)3-s − 0.925i·7-s − 0.999i·9-s − 1.47·11-s + 0.554·13-s − 1.45i·17-s + 1.12i·19-s + (−0.654 − 0.654i)21-s − 0.510·23-s + (−0.707 − 0.707i)27-s − 1.75i·31-s + (−1.04 + 1.04i)33-s − 0.328·37-s + (0.392 − 0.392i)39-s + 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543082717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543082717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.79iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 7.34iT - 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 4.89iT - 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.603073229259447999192073614406, −8.247613946220702407669197795105, −7.917417206527843877225574932805, −7.15753487592582900396557740391, −6.25432680947147746309612162174, −5.20626979728758973302587263313, −4.01146614273138587152380748832, −3.08696525527140770893926355734, −2.03004059375596341958878570756, −0.57306524434504561419056415933,
2.00752560891054778093748432098, 2.88596053676174090621398366851, 3.84662769459138831875254747179, 5.03121135831868794505126131594, 5.57051271290936632965492023754, 6.76978532143042226733252310235, 7.947539952930556073604185462880, 8.520166169847606950564519837793, 9.043820266590559406152081275947, 10.18487641930995925981340910571
