| L(s) = 1 | + 43.8·5-s − 238i·13-s − 111. i·17-s + 1.29e3·25-s − 1.12e3·29-s − 1.68e3i·37-s − 3.16e3i·41-s − 2.40e3·49-s − 5.31e3·53-s − 2.64e3i·61-s − 1.04e4i·65-s + 1.05e4·73-s − 4.89e3i·85-s + 1.57e4i·89-s − 1.87e4·97-s + ⋯ |
| L(s) = 1 | + 1.75·5-s − 1.40i·13-s − 0.386i·17-s + 2.07·25-s − 1.34·29-s − 1.22i·37-s − 1.88i·41-s − 49-s − 1.89·53-s − 0.709i·61-s − 2.46i·65-s + 1.98·73-s − 0.677i·85-s + 1.98i·89-s − 1.98·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.508939248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508939248\) |
| \(L(3)\) |
|
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 | \( 1 - 43.8T + 625T^{2} \) |
| 7 | \( 1 + 2.40e3T^{2} \) |
| 11 | \( 1 + 1.46e4T^{2} \) |
| 13 | \( 1 + 238iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 111. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.12e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.68e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 3.16e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.31e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.64e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.05e4T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.57e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.87e4T + 8.85e7T^{2} \) |
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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
−9.334303170816926082207141661465, −8.254762107864174725493470411396, −7.31203709314740746042216143264, −6.33561218054273820426399519917, −5.56488050194677280796988217088, −5.09732222260479878408160935313, −3.57083085517741561393896632546, −2.52457870166914904805372426669, −1.69164685527601505982600356270, −0.45335212681523934586054292632,
1.40848708688443499730356699667, 1.97131160794564611997223165310, 3.10363804760620292875707558210, 4.44959700134859732826294515962, 5.28401953386302939380255766921, 6.29464238575916581913667246602, 6.60730260311598004187906106571, 7.86542254953108736696644672544, 8.928291921379824980494037922975, 9.529783657856649961702318071475
