| L(s) = 1 | + 3i·3-s + 4i·7-s − 9·9-s + 48·11-s − 2i·13-s − 114i·17-s + 140·19-s − 12·21-s + 72i·23-s − 27i·27-s − 210·29-s − 272·31-s + 144i·33-s − 334i·37-s + 6·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.215i·7-s − 0.333·9-s + 1.31·11-s − 0.0426i·13-s − 1.62i·17-s + 1.69·19-s − 0.124·21-s + 0.652i·23-s − 0.192i·27-s − 1.34·29-s − 1.57·31-s + 0.759i·33-s − 1.48i·37-s + 0.0246·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.063733399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063733399\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 - 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 114iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 140T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 210T + 2.43e4T^{2} \) |
| 31 | \( 1 + 272T + 2.97e4T^{2} \) |
| 37 | \( 1 + 334iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 216iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 78iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 240T + 2.05e5T^{2} \) |
| 61 | \( 1 - 302T + 2.26e5T^{2} \) |
| 67 | \( 1 + 596iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 768T + 3.57e5T^{2} \) |
| 73 | \( 1 - 478iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 640T + 4.93e5T^{2} \) |
| 83 | \( 1 + 348iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 210T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3iT - 9.12e5T^{2} \) |
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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
−9.312330133089882012630771800149, −8.871559736343740868445159615325, −7.44456820201042208779106791273, −7.05710802281960569950249482998, −5.63641496991596172262758155199, −5.23391451551795439624459672109, −3.92338759499866711659654893239, −3.31210587949333384787610363608, −1.94103194222554993707145549004, −0.56398853839584598481768331207,
1.05292629844596158041362267066, 1.83957999922518805068485831117, 3.32786197715328077176298711000, 4.05721486601111057134018429552, 5.34628656762851323706343060284, 6.19560886629945639039652644543, 6.94910197208657226420546753877, 7.72943548797226142481689761762, 8.605876664799966058516507915818, 9.359799366833261840636254597388
