L(s) = 1 | + (−0.350 − 0.350i)3-s + 3.08i·5-s + (−0.707 − 0.707i)7-s − 2.75i·9-s + (1.14 + 1.14i)11-s + (−2.88 − 2.88i)13-s + (1.08 − 1.08i)15-s + (3.58 − 3.58i)17-s + (2.29 − 2.29i)19-s + 0.495i·21-s + 3.46·23-s − 4.51·25-s + (−2.01 + 2.01i)27-s + (4.60 + 4.60i)29-s + 2.08·31-s + ⋯ |
L(s) = 1 | + (−0.202 − 0.202i)3-s + 1.37i·5-s + (−0.267 − 0.267i)7-s − 0.917i·9-s + (0.344 + 0.344i)11-s + (−0.799 − 0.799i)13-s + (0.279 − 0.279i)15-s + (0.869 − 0.869i)17-s + (0.526 − 0.526i)19-s + 0.108i·21-s + 0.723·23-s − 0.902·25-s + (−0.388 + 0.388i)27-s + (0.855 + 0.855i)29-s + 0.374·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.481845347\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481845347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (-6.11 + 1.90i)T \) |
good | 3 | \( 1 + (0.350 + 0.350i)T + 3iT^{2} \) |
| 5 | \( 1 - 3.08iT - 5T^{2} \) |
| 11 | \( 1 + (-1.14 - 1.14i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.88 + 2.88i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.58 + 3.58i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.29 + 2.29i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + (-4.60 - 4.60i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.08T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 43 | \( 1 - 6.95iT - 43T^{2} \) |
| 47 | \( 1 + (3.66 - 3.66i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.69 - 5.69i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.42T + 59T^{2} \) |
| 61 | \( 1 + 8.24iT - 61T^{2} \) |
| 67 | \( 1 + (5.59 - 5.59i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.14 + 3.14i)T + 71iT^{2} \) |
| 73 | \( 1 + 8.57iT - 73T^{2} \) |
| 79 | \( 1 + (-5.97 - 5.97i)T + 79iT^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (3.25 + 3.25i)T + 89iT^{2} \) |
| 97 | \( 1 + (3.75 - 3.75i)T - 97iT^{2} \) |
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
−9.763353236707743114320991085976, −9.282972552065485792645820290381, −7.82164475495883265203467862577, −7.14811401163243429961649147981, −6.64778617400298141142950452109, −5.71698613560483897996589304464, −4.58413432564665620325565462883, −3.18700655625468249838749600001, −2.84109396342901266721127242815, −0.872932730752407173447328974103,
1.05989042450555251818808440981, 2.35370309808096976076237064151, 3.86659494852933506254092164430, 4.75246096651442764932470783789, 5.41480087048691430364516903357, 6.27956895342253662828820136099, 7.54625848430606562155884080851, 8.258818182517275701474151759463, 9.010711402451366627397441232983, 9.770000852009363715801324980440