| L(s) = 1 | + 5.19·3-s + 30.1i·5-s + 52.3i·7-s + 27·9-s − 90.0·11-s + 60.3i·13-s + 156. i·15-s − 338·17-s − 6.92·19-s + 271. i·21-s + 732. i·23-s − 287·25-s + 140.·27-s − 1.29e3i·29-s − 1.30e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20i·5-s + 1.06i·7-s + 0.333·9-s − 0.744·11-s + 0.357i·13-s + 0.697i·15-s − 1.16·17-s − 0.0191·19-s + 0.616i·21-s + 1.38i·23-s − 0.459·25-s + 0.192·27-s − 1.54i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.179040051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179040051\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19T \) |
| good | 5 | \( 1 - 30.1iT - 625T^{2} \) |
| 7 | \( 1 - 52.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 90.0T + 1.46e4T^{2} \) |
| 13 | \( 1 - 60.3iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 338T + 8.35e4T^{2} \) |
| 19 | \( 1 + 6.92T + 1.30e5T^{2} \) |
| 23 | \( 1 - 732. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.29e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 241. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 578T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.02e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.44e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.19e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 6.40e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 8.26e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.28e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.12e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.31e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 910T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.42e3T + 8.85e7T^{2} \) |
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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
−11.25748202103208226957781107201, −10.16471786252634583780990598609, −9.377939581028279626815113328845, −8.388571076779418360897787231418, −7.46166024684715774792032001736, −6.49968589366631493045522382732, −5.49907790169617691126044761718, −4.03293527824268505358525342081, −2.76306106360925704755386886234, −2.13691012841414052899109094456,
0.29137577582836413102053743548, 1.51484997005490701423547336672, 3.04755833984427561508606737221, 4.40882115644136230983390299240, 5.00862572319347721066005396858, 6.58416583268060051005450190116, 7.58949365398757466259232689807, 8.524213033739391653375392252371, 9.093274120683510103288671711946, 10.35990455923741024599681596368
