L(s) = 1 | + (0.258 − 0.965i)3-s + (−1.15 − 2.38i)7-s + (−0.866 − 0.499i)9-s + (−1.36 − 2.36i)11-s + (−0.707 − 0.707i)13-s + (6.50 + 1.74i)17-s + (4.23 − 7.33i)19-s + (−2.59 + 0.500i)21-s + (1.93 + 7.20i)23-s + (−0.707 + 0.707i)27-s − 6.92i·29-s + (−1.73 + i)31-s + (−2.63 + 0.707i)33-s + (−10.8 + 2.89i)37-s + (−0.866 + 0.500i)39-s + ⋯ |
L(s) = 1 | + (0.149 − 0.557i)3-s + (−0.436 − 0.899i)7-s + (−0.288 − 0.166i)9-s + (−0.411 − 0.713i)11-s + (−0.196 − 0.196i)13-s + (1.57 + 0.422i)17-s + (0.970 − 1.68i)19-s + (−0.566 + 0.109i)21-s + (0.402 + 1.50i)23-s + (−0.136 + 0.136i)27-s − 1.28i·29-s + (−0.311 + 0.179i)31-s + (−0.459 + 0.123i)33-s + (−1.77 + 0.476i)37-s + (−0.138 + 0.0800i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.250313708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250313708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.15 + 2.38i)T \) |
good | 11 | \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (-6.50 - 1.74i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.23 + 7.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.93 - 7.20i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + (1.73 - i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (10.8 - 2.89i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.19iT - 41T^{2} \) |
| 43 | \( 1 + (0.378 - 0.378i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.63 + 9.84i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (8.43 + 2.26i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (4.63 + 8.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.42 + 4.86i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.81 + 6.76i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (3.98 - 14.8i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.96 - 1.13i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.34 + 3.34i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.63 + 6.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.50 + 2.50i)T - 97iT^{2} \) |
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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
−8.704267726108446278977532872681, −7.76693940342747100189929814471, −7.39247082439324856352465451099, −6.54315320690757390459093713159, −5.61081951681273896777995623774, −4.88537629750441634198934327551, −3.38325956234557770621002199820, −3.16625780137755999327350384651, −1.53518587654110092093451179492, −0.43661321582566650251064518383,
1.61860235675289328476401485984, 2.88119116509988824600101034569, 3.47799716979080653263062776663, 4.72401031589193785149321426090, 5.40832471598858605363957372920, 6.08693759298869467785187889846, 7.25074849946347463488443252947, 7.86302271931072363037325723496, 8.855798621225761940568470920072, 9.371893021195597005393790951270