| L(s) = 1 | − 44.0i·5-s − 31·7-s + 220. i·11-s − 241·13-s − 220. i·17-s − 271·19-s − 220. i·23-s − 1.31e3·25-s − 440. i·29-s − 778·31-s + 1.36e3i·35-s + 1.07e3·37-s + 2.20e3i·41-s − 298·43-s − 3.30e3i·47-s + ⋯ |
| L(s) = 1 | − 1.76i·5-s − 0.632·7-s + 1.82i·11-s − 1.42·13-s − 0.762i·17-s − 0.750·19-s − 0.416i·23-s − 2.11·25-s − 0.524i·29-s − 0.809·31-s + 1.11i·35-s + 0.788·37-s + 1.31i·41-s − 0.161·43-s − 1.49i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(-0.453635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.453635i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 44.0iT - 625T^{2} \) |
| 7 | \( 1 + 31T + 2.40e3T^{2} \) |
| 11 | \( 1 - 220. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 241T + 2.85e4T^{2} \) |
| 17 | \( 1 + 220. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 271T + 1.30e5T^{2} \) |
| 23 | \( 1 + 220. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 440. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 778T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.07e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.20e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 298T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.30e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.08e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.86e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.64e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.60e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.40e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.19e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 329T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.32e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.15e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.59e4T + 8.85e7T^{2} \) |
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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
−12.64836593810656607525886313572, −11.81010888749555906893882801025, −9.812538473002807956575916533835, −9.493627489051444843114398073699, −8.104689169798971206088529223301, −6.90498690418368517750519345952, −5.13327308327571124320268155979, −4.41114827908719871320249860904, −2.08148830593690100255799207609, −0.18773472078952524794597144502,
2.62287391120781756648429351322, 3.62080101515407406196007772047, 5.83763632179216992590037879961, 6.72334436103308784383730694449, 7.85169865390757920266217397772, 9.375581766525475270988699124412, 10.58230208507317201608599990142, 11.09697444729623078370047166097, 12.45022564739218180290274335378, 13.71823502560815250954889667171
