L(s) = 1 | + (1.11 + 0.646i)5-s + 0.567·7-s − 5.19i·11-s + (−6.03 + 3.48i)13-s + (0.318 + 0.184i)17-s + (−1.16 − 4.20i)19-s + (−6.57 + 3.79i)23-s + (−1.66 − 2.88i)25-s + (3.70 + 6.41i)29-s + 6.51i·31-s + (0.635 + 0.366i)35-s + 4.89i·37-s + (−5.62 + 9.73i)41-s + (2.44 − 4.24i)43-s + (−7.37 + 4.25i)47-s + ⋯ |
L(s) = 1 | + (0.500 + 0.288i)5-s + 0.214·7-s − 1.56i·11-s + (−1.67 + 0.966i)13-s + (0.0773 + 0.0446i)17-s + (−0.267 − 0.963i)19-s + (−1.37 + 0.791i)23-s + (−0.332 − 0.576i)25-s + (0.687 + 1.19i)29-s + 1.17i·31-s + (0.107 + 0.0619i)35-s + 0.805i·37-s + (−0.877 + 1.52i)41-s + (0.373 − 0.646i)43-s + (−1.07 + 0.621i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5344023034\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5344023034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.16 + 4.20i)T \) |
good | 5 | \( 1 + (-1.11 - 0.646i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 0.567T + 7T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + (6.03 - 3.48i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.318 - 0.184i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (6.57 - 3.79i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.70 - 6.41i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.51iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + (5.62 - 9.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.44 + 4.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.37 - 4.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.03 - 5.26i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.03 - 5.26i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.52 - 2.64i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.10 - 0.640i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.67 + 6.36i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.43 + 12.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.12 + 4.69i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.3iT - 83T^{2} \) |
| 89 | \( 1 + (-0.953 - 1.65i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.75 - 2.16i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.132567769176304142621869233974, −8.413028166037182249471754097467, −7.65203974558504799070243842578, −6.68565315684817243673451144443, −6.23848366475231711975286731230, −5.17582430997142572981903570689, −4.58783578689024375302607959057, −3.34669610774603291239285791331, −2.58096545218896602730240709614, −1.50087487241476900323120154191,
0.15651055491135071842195797082, 1.94756859782022455657532169444, 2.35693688457022126255385900452, 3.83911599769943496993888251710, 4.66926637028696286771358762483, 5.33954149534499229656509722630, 6.13781848653337245639040500562, 7.12933761046186702461970648426, 7.78894767493502375254905242960, 8.335311610150197794196122795489