L(s) = 1 | + (0.904 − 1.08i)2-s + (−1.78 − 1.78i)3-s + (−0.364 − 1.96i)4-s + (−1.34 + 1.78i)5-s + (−3.56 + 0.327i)6-s + (−2.34 + 2.34i)7-s + (−2.46 − 1.38i)8-s + 3.39i·9-s + (0.733 + 3.07i)10-s − 5.90i·11-s + (−2.86 + 4.16i)12-s + (−0.707 + 0.707i)13-s + (0.429 + 4.66i)14-s + (5.59 − 0.803i)15-s + (−3.73 + 1.43i)16-s + (−0.975 − 0.975i)17-s + ⋯ |
L(s) = 1 | + (0.639 − 0.768i)2-s + (−1.03 − 1.03i)3-s + (−0.182 − 0.983i)4-s + (−0.599 + 0.800i)5-s + (−1.45 + 0.133i)6-s + (−0.886 + 0.886i)7-s + (−0.872 − 0.488i)8-s + 1.13i·9-s + (0.232 + 0.972i)10-s − 1.78i·11-s + (−0.826 + 1.20i)12-s + (−0.196 + 0.196i)13-s + (0.114 + 1.24i)14-s + (1.44 − 0.207i)15-s + (−0.933 + 0.358i)16-s + (−0.236 − 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.162183 + 0.450341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.162183 + 0.450341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.904 + 1.08i)T \) |
| 5 | \( 1 + (1.34 - 1.78i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.78 + 1.78i)T + 3iT^{2} \) |
| 7 | \( 1 + (2.34 - 2.34i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.90iT - 11T^{2} \) |
| 17 | \( 1 + (0.975 + 0.975i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + (-3.10 - 3.10i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.56iT - 29T^{2} \) |
| 31 | \( 1 + 7.23iT - 31T^{2} \) |
| 37 | \( 1 + (2.19 + 2.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.538T + 41T^{2} \) |
| 43 | \( 1 + (5.78 + 5.78i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.98 - 3.98i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.09 + 7.09i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + (2.58 - 2.58i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.53iT - 71T^{2} \) |
| 73 | \( 1 + (-2.76 + 2.76i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 + (-6.93 - 6.93i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.42iT - 89T^{2} \) |
| 97 | \( 1 + (11.8 + 11.8i)T + 97iT^{2} \) |
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
−11.42683233221069648431565721059, −11.13640197879561708272169862258, −9.817539017324087237972705579298, −8.513875789135884433665014488700, −6.95681993258115120626980838626, −6.16717073106076472783486154103, −5.54492410276289851383778346278, −3.66175644745402116693559563515, −2.50214223503006783684390585541, −0.33059063743270208101684517845,
3.63623084899759788770557214643, 4.60542405179351538907295672878, 5.05507920846573036862181003566, 6.57023668165043660673345970620, 7.29697782824314197050719744904, 8.713947178194216152595169703785, 9.814620084008887268724067466861, 10.62923748511139522450921026967, 11.88019471795216459542470081969, 12.59357660136679172392980443519