| L(s) = 1 | − 0.396i·2-s − 11.4i·3-s + 31.8·4-s − 4.55·6-s − 8.04i·7-s − 25.3i·8-s + 110.·9-s + 335.·11-s − 366. i·12-s − 169i·13-s − 3.19·14-s + 1.00e3·16-s − 38.1i·17-s − 43.9i·18-s − 1.17e3·19-s + ⋯ |
| L(s) = 1 | − 0.0701i·2-s − 0.737i·3-s + 0.995·4-s − 0.0517·6-s − 0.0620i·7-s − 0.139i·8-s + 0.456·9-s + 0.837·11-s − 0.733i·12-s − 0.277i·13-s − 0.00435·14-s + 0.985·16-s − 0.0320i·17-s − 0.0319i·18-s − 0.749·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.172682102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.172682102\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 + 0.396iT - 32T^{2} \) |
| 3 | \( 1 + 11.4iT - 243T^{2} \) |
| 7 | \( 1 + 8.04iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 335.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 38.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.67e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.71e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.63e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.95e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.17e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.17e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.61e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.75e5iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62791589917596542648521873950, −9.896543809198597510729745232787, −8.512710830931869934540923669104, −7.57843028752942039430398243580, −6.69808431864783242605252017753, −6.12765866893591834816283564863, −4.49622016377348716970923409345, −3.09650945543894475618184885924, −1.86540403610743283646628988876, −0.940284335217437725910994376327,
1.21033263059149027270926338228, 2.53690896606775342189793642960, 3.84479867929998221671243409071, 4.82441467496599727968044140728, 6.28330625443407165857755591769, 6.86467150418391950247285975724, 8.130563526767445718772365978761, 9.154831279245133542781750289175, 10.24081113373247996810230521309, 10.72571184267423537906986321940
