| L(s) = 1 | − 22i·5-s − 92i·13-s − 104·17-s − 359·25-s + 130i·29-s − 396i·37-s + 472·41-s − 343·49-s + 518i·53-s − 468i·61-s − 2.02e3·65-s + 1.09e3·73-s + 2.28e3i·85-s + 176·89-s + 594·97-s + ⋯ |
| L(s) = 1 | − 1.96i·5-s − 1.96i·13-s − 1.48·17-s − 2.87·25-s + 0.832i·29-s − 1.75i·37-s + 1.79·41-s − 49-s + 1.34i·53-s − 0.982i·61-s − 3.86·65-s + 1.76·73-s + 2.91i·85-s + 0.209·89-s + 0.621·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8641770387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641770387\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 22iT - 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 92iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 104T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 130iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 396iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 472T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 518iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5T^{2} \) |
| 61 | \( 1 + 468iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 - 176T + 7.04e5T^{2} \) |
| 97 | \( 1 - 594T + 9.12e5T^{2} \) |
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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
−8.970006115422341557208766988141, −8.155232807849429526434579756431, −7.56244373268233956676761075543, −6.12461171988710701978683178881, −5.35741314783722304954906670220, −4.70755978975232273079404152065, −3.75637651333367935481440153593, −2.32124045454665610269267548388, −1.03343237334104771516633581593, −0.22373686823321245993206599400,
1.93899500077636139424083535611, 2.63577257335708424965134835131, 3.78605295307496528134774519253, 4.56623665985779360221896090138, 6.16727916950176754144370955983, 6.61978373304166425076575475314, 7.18321009563188328316342222284, 8.209041411040670357141517541946, 9.318558783853312381684583871259, 9.909413659314296968398052073323
