| L(s) = 1 | − i·2-s + 3-s − 4-s − 2.60i·5-s − i·6-s + i·8-s + 9-s − 2.60·10-s + 3.60i·11-s − 12-s + (3.60 − 0.167i)13-s − 2.60i·15-s + 16-s − 2.83·17-s − i·18-s + 1.33i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.16i·5-s − 0.408i·6-s + 0.353i·8-s + 0.333·9-s − 0.822·10-s + 1.08i·11-s − 0.288·12-s + (0.998 − 0.0463i)13-s − 0.671i·15-s + 0.250·16-s − 0.687·17-s − 0.235i·18-s + 0.306i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.397554241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.397554241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-3.60 + 0.167i)T \) |
| good | 5 | \( 1 + 2.60iT - 5T^{2} \) |
| 11 | \( 1 - 3.60iT - 11T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 1.33iT - 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 0.167iT - 37T^{2} \) |
| 41 | \( 1 + 3.03iT - 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 9.74iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.76iT - 71T^{2} \) |
| 73 | \( 1 + 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 8.10iT - 83T^{2} \) |
| 89 | \( 1 - 8.16iT - 89T^{2} \) |
| 97 | \( 1 - 0.139iT - 97T^{2} \) |
| 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
−8.468701192789382449595218969400, −7.911675070797270295017426858935, −6.91721474553035918054008449602, −6.01231524700298015968580921199, −4.90412269626602145263989123583, −4.49487035941665015802405928684, −3.69200207706962225841818586844, −2.63119354034317188321171290113, −1.71826143142107740108311918509, −0.846682846037226602180884380396,
0.994312977097586606561374156076, 2.53616617856832637946742459849, 3.22806789228018772982611396017, 3.94839230540273969561276907570, 4.97250738676031974703919328658, 5.94990809493656569269535817860, 6.65584507873562688369639519190, 6.98498938164988055022005027585, 8.002740938156519455805810761296, 8.636594387187166732855381099636
