| L(s) = 1 | − 3-s + 9i·5-s + 15i·7-s − 26·9-s + 48i·11-s + (−26 + 39i)13-s − 9i·15-s − 45·17-s + 6i·19-s − 15i·21-s − 162·23-s + 44·25-s + 53·27-s + 144·29-s − 264i·31-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.804i·5-s + 0.809i·7-s − 0.962·9-s + 1.31i·11-s + (−0.554 + 0.832i)13-s − 0.154i·15-s − 0.642·17-s + 0.0724i·19-s − 0.155i·21-s − 1.46·23-s + 0.351·25-s + 0.377·27-s + 0.922·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3830582727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3830582727\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (26 - 39i)T \) |
| good | 3 | \( 1 + T + 27T^{2} \) |
| 5 | \( 1 - 9iT - 125T^{2} \) |
| 7 | \( 1 - 15iT - 343T^{2} \) |
| 11 | \( 1 - 48iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 45T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 162T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 303iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 192iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 97T + 7.95e4T^{2} \) |
| 47 | \( 1 - 111iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 414T + 1.48e5T^{2} \) |
| 59 | \( 1 + 522iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 376T + 2.26e5T^{2} \) |
| 67 | \( 1 + 36iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 357iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 830T + 4.93e5T^{2} \) |
| 83 | \( 1 + 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 438iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 852iT - 9.12e5T^{2} \) |
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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
−10.29474357091337669498022399551, −9.617459574774060378787596515001, −8.733835200973176983468176123280, −7.80467048937980274417062171219, −6.78147894958442080186264630825, −6.18425214573368360273020194862, −5.09338992440553541879344352040, −4.12773472379395826103322412452, −2.66903502871139819040313725781, −2.08025333831088413047392222568,
0.11934305292462344514840519436, 0.966812582967932117317803753716, 2.67238809187542473917886240230, 3.74433153962169803426200218084, 4.88025474610527058113845566936, 5.65916083046066507649657554575, 6.53759941029997810571346110574, 7.73501707051363714235749851869, 8.489244156769096636092549120081, 9.046531235905325038624886925539
