L(s) = 1 | + (−1.38 + 1.38i)3-s − 2.11i·5-s + (0.707 − 0.707i)7-s − 0.852i·9-s + (−0.520 + 0.520i)11-s + (0.612 − 0.612i)13-s + (2.93 + 2.93i)15-s + (1.16 + 1.16i)17-s + (−2.68 − 2.68i)19-s + 1.96i·21-s − 3.65·23-s + 0.540·25-s + (−2.98 − 2.98i)27-s + (3.93 − 3.93i)29-s − 4.96·31-s + ⋯ |
L(s) = 1 | + (−0.801 + 0.801i)3-s − 0.944i·5-s + (0.267 − 0.267i)7-s − 0.284i·9-s + (−0.156 + 0.156i)11-s + (0.169 − 0.169i)13-s + (0.756 + 0.756i)15-s + (0.281 + 0.281i)17-s + (−0.616 − 0.616i)19-s + 0.428i·21-s − 0.762·23-s + 0.108·25-s + (−0.573 − 0.573i)27-s + (0.730 − 0.730i)29-s − 0.891·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8575038752\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8575038752\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.38 + 6.25i)T \) |
good | 3 | \( 1 + (1.38 - 1.38i)T - 3iT^{2} \) |
| 5 | \( 1 + 2.11iT - 5T^{2} \) |
| 11 | \( 1 + (0.520 - 0.520i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.612 + 0.612i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.16 - 1.16i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.68 + 2.68i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + (-3.93 + 3.93i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 43 | \( 1 + 8.44iT - 43T^{2} \) |
| 47 | \( 1 + (1.57 + 1.57i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.25 + 2.25i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.36T + 59T^{2} \) |
| 61 | \( 1 + 8.42iT - 61T^{2} \) |
| 67 | \( 1 + (0.439 + 0.439i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.13 - 1.13i)T - 71iT^{2} \) |
| 73 | \( 1 + 0.278iT - 73T^{2} \) |
| 79 | \( 1 + (3.94 - 3.94i)T - 79iT^{2} \) |
| 83 | \( 1 - 8.17T + 83T^{2} \) |
| 89 | \( 1 + (-11.2 + 11.2i)T - 89iT^{2} \) |
| 97 | \( 1 + (9.32 + 9.32i)T + 97iT^{2} \) |
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
−9.800158626306665676953535962841, −8.811367111257089805094754337484, −8.172302254084805064277420056856, −7.11465881617390618117042861141, −5.95957134240459167478928727451, −5.23485088977041695913625320003, −4.55293107322971715480106504299, −3.77458784523937417360888536849, −2.04668086402201015575524145590, −0.43880591624464557160992426087,
1.33649369156331754291351761632, 2.59849216362833776135793006691, 3.73605871886615701978475744411, 5.05550851432952151008188543466, 6.03631115308922574059680622086, 6.52609208233857573568683930830, 7.35431904422556831189920728397, 8.126885131060409619458864309224, 9.157460173787537606816474804470, 10.20970870429803880942912890531