L(s) = 1 | + (1.07 + 1.95i)5-s + (2.36 + 2.36i)7-s + 1.43i·11-s + (−4.27 + 4.27i)13-s + (−0.739 + 0.739i)17-s − 3.16i·19-s + (−0.707 − 0.707i)23-s + (−2.67 + 4.22i)25-s + 8.01·29-s + 7.15·31-s + (−2.08 + 7.16i)35-s + (−6.14 − 6.14i)37-s + 10.2i·41-s + (−1.42 + 1.42i)43-s + (−5.93 + 5.93i)47-s + ⋯ |
L(s) = 1 | + (0.481 + 0.876i)5-s + (0.892 + 0.892i)7-s + 0.432i·11-s + (−1.18 + 1.18i)13-s + (−0.179 + 0.179i)17-s − 0.726i·19-s + (−0.147 − 0.147i)23-s + (−0.535 + 0.844i)25-s + 1.48·29-s + 1.28·31-s + (−0.351 + 1.21i)35-s + (−1.01 − 1.01i)37-s + 1.60i·41-s + (−0.217 + 0.217i)43-s + (−0.865 + 0.865i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.786655223\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786655223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.07 - 1.95i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-2.36 - 2.36i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 + (4.27 - 4.27i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.739 - 0.739i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 + (6.14 + 6.14i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (1.42 - 1.42i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.93 - 5.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.27 - 4.27i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + (2.37 + 2.37i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.71iT - 71T^{2} \) |
| 73 | \( 1 + (9.82 - 9.82i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.7iT - 79T^{2} \) |
| 83 | \( 1 + (-5.86 - 5.86i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 + (9.11 + 9.11i)T + 97iT^{2} \) |
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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
−8.704804683520387001420471490621, −7.988112480545554492115151428293, −7.09498793568438494046438132226, −6.60895688600649898553650919510, −5.82025510181791287368448709965, −4.78390921993559166815685358709, −4.50111345180651870319780198295, −2.96193835138765807522569703459, −2.38040485000772094003229845686, −1.60923273232518866117490752975,
0.49262360259629989552305493869, 1.40028116691429600940149154525, 2.47133635614800863891934056192, 3.54654654622554493405817257983, 4.61295689080535167739430916708, 5.00951526001087476476944906438, 5.74717429214751943173601853029, 6.71983625621157385508640769463, 7.53738016334711797060514165076, 8.281417216695546054080705705951