L(s) = 1 | + (1.19 + 0.756i)2-s + (1.42 − 1.42i)3-s + (0.854 + 1.80i)4-s + (0.702 + 0.702i)5-s + (2.78 − 0.625i)6-s + (−0.348 + 2.80i)8-s − 1.08i·9-s + (0.307 + 1.37i)10-s + (−1.38 − 1.38i)11-s + (3.80 + 1.36i)12-s + (2.10 − 2.10i)13-s + 2.00·15-s + (−2.54 + 3.08i)16-s + 5.67·17-s + (0.822 − 1.29i)18-s + (0.451 − 0.451i)19-s + ⋯ |
L(s) = 1 | + (0.844 + 0.535i)2-s + (0.825 − 0.825i)3-s + (0.427 + 0.904i)4-s + (0.313 + 0.313i)5-s + (1.13 − 0.255i)6-s + (−0.123 + 0.992i)8-s − 0.362i·9-s + (0.0971 + 0.433i)10-s + (−0.416 − 0.416i)11-s + (1.09 + 0.393i)12-s + (0.583 − 0.583i)13-s + 0.518·15-s + (−0.635 + 0.772i)16-s + 1.37·17-s + (0.193 − 0.306i)18-s + (0.103 − 0.103i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.32259 + 0.830406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.32259 + 0.830406i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 - 0.756i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.42 + 1.42i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.702 - 0.702i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.38 + 1.38i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 + (-0.451 + 0.451i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.84iT - 23T^{2} \) |
| 29 | \( 1 + (0.207 - 0.207i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + (7.03 + 7.03i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.40iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 + 0.289T + 47T^{2} \) |
| 53 | \( 1 + (5.58 + 5.58i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.84 + 9.84i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.64 + 2.64i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.35 - 7.35i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 0.358iT - 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + (-0.424 + 0.424i)T - 83iT^{2} \) |
| 89 | \( 1 + 17.5iT - 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.50417429549229699405725373663, −9.244923773805274765071020327928, −8.245165035622159157588599640237, −7.70031465823484582712481321693, −7.01995386159276081206648150982, −5.86774900136331464097340875891, −5.29467200530568184274948350301, −3.62871707598285840631228155492, −2.98997025320225414827683745059, −1.77985338165153939733378736468,
1.55745683715994428157896525998, 2.88896645227206919731104808131, 3.70138448381045808032196846264, 4.61618056074856154302852545336, 5.47545246222642784615456355152, 6.51040804156941449459599816069, 7.70021229340976863571126524478, 8.906953578138277833941494643726, 9.456663126507585907707061169758, 10.29813874430930653152788221529