L(s) = 1 | + (0.707 − 0.707i)3-s + (1 + i)5-s + 2.82i·7-s − 1.00i·9-s + (−3 + 3i)13-s + 1.41·15-s − 4·17-s + (−5.65 + 5.65i)19-s + (2.00 + 2.00i)21-s + 5.65i·23-s − 3i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 2.82·31-s + (−2.82 + 2.82i)35-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.447 + 0.447i)5-s + 1.06i·7-s − 0.333i·9-s + (−0.832 + 0.832i)13-s + 0.365·15-s − 0.970·17-s + (−1.29 + 1.29i)19-s + (0.436 + 0.436i)21-s + 1.17i·23-s − 0.600i·25-s + (−0.136 − 0.136i)27-s + (0.185 − 0.185i)29-s − 0.508·31-s + (−0.478 + 0.478i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320228017\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320228017\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - i)T + 5iT^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + (5.65 - 5.65i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + (-1 + i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (8.48 + 8.48i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−9.524181587881883309779907812938, −8.954502589648945993896177563123, −8.182332956821665188916979337120, −7.27131882533466088449557110451, −6.40390802968392129225036680886, −5.85697096347774388926050079889, −4.70689049039058344467916473538, −3.62589579254904414616495942946, −2.31454934439726561708396904633, −1.97973961080842899140963452601,
0.44481497311318356182866803924, 2.07217648548690090654631181573, 3.06410157887943608936931090452, 4.38776690526857730509386779569, 4.70459345029056895069463175659, 5.88477496103395478460500536407, 6.95882742480187433864895793284, 7.52351599362164634515170297258, 8.744236046474594522246002025660, 8.981347940714815704412372192826