L(s) = 1 | − 1.76e4·5-s − 1.59e6·9-s − 7.25e6·13-s − 8.58e6·17-s − 9.08e8·25-s + 1.48e9·29-s + 2.61e10·37-s − 4.83e10·41-s + 2.81e10·45-s − 9.68e10·49-s + 2.86e11·53-s + 3.08e11·61-s + 1.28e11·65-s + 2.58e12·73-s + 2.54e12·81-s + 1.51e11·85-s + 7.78e12·89-s + 1.08e13·97-s + 2.05e13·101-s + 2.11e13·109-s − 3.53e12·113-s + 1.15e13·117-s + ⋯ |
L(s) = 1 | − 0.505·5-s − 9-s − 0.416·13-s − 0.0862·17-s − 0.743·25-s + 0.464·29-s + 1.67·37-s − 1.58·41-s + 0.505·45-s − 49-s + 1.77·53-s + 0.766·61-s + 0.210·65-s + 1.99·73-s + 81-s + 0.0436·85-s + 1.65·89-s + 1.32·97-s + 1.92·101-s + 1.20·109-s − 0.159·113-s + 0.416·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(1.272959675\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272959675\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + p^{13} T^{2} \) |
| 5 | \( 1 + 17678 T + p^{13} T^{2} \) |
| 7 | \( 1 + p^{13} T^{2} \) |
| 11 | \( 1 + p^{13} T^{2} \) |
| 13 | \( 1 + 7254006 T + p^{13} T^{2} \) |
| 17 | \( 1 + 8582078 T + p^{13} T^{2} \) |
| 19 | \( 1 + p^{13} T^{2} \) |
| 23 | \( 1 + p^{13} T^{2} \) |
| 29 | \( 1 - 1486672730 T + p^{13} T^{2} \) |
| 31 | \( 1 + p^{13} T^{2} \) |
| 37 | \( 1 - 26174743922 T + p^{13} T^{2} \) |
| 41 | \( 1 + 48310539670 T + p^{13} T^{2} \) |
| 43 | \( 1 + p^{13} T^{2} \) |
| 47 | \( 1 + p^{13} T^{2} \) |
| 53 | \( 1 - 286798198946 T + p^{13} T^{2} \) |
| 59 | \( 1 + p^{13} T^{2} \) |
| 61 | \( 1 - 308331475930 T + p^{13} T^{2} \) |
| 67 | \( 1 + p^{13} T^{2} \) |
| 71 | \( 1 + p^{13} T^{2} \) |
| 73 | \( 1 - 2582975771994 T + p^{13} T^{2} \) |
| 79 | \( 1 + p^{13} T^{2} \) |
| 83 | \( 1 + p^{13} T^{2} \) |
| 89 | \( 1 - 7780304165930 T + p^{13} T^{2} \) |
| 97 | \( 1 - 10874684920338 T + p^{13} T^{2} \) |
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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
−12.01450317146701149363837327357, −11.28370655925478330737102044733, −9.952816548196404857177916642762, −8.667068551364035388868341314754, −7.68065145391708285263636049957, −6.28777913814001257874068942568, −4.99450360259779746057580373494, −3.59295171318543287746668332850, −2.31839339534315798404063901230, −0.57077185902908256993431499083,
0.57077185902908256993431499083, 2.31839339534315798404063901230, 3.59295171318543287746668332850, 4.99450360259779746057580373494, 6.28777913814001257874068942568, 7.68065145391708285263636049957, 8.667068551364035388868341314754, 9.952816548196404857177916642762, 11.28370655925478330737102044733, 12.01450317146701149363837327357