L(s) = 1 | + (0.857 − 1.12i)2-s + (−0.416 + 0.416i)3-s + (−0.529 − 1.92i)4-s + (1.13 + 1.13i)5-s + (0.111 + 0.826i)6-s + (−2.62 − 1.05i)8-s + 2.65i·9-s + (2.24 − 0.302i)10-s + (3.85 + 3.85i)11-s + (1.02 + 0.583i)12-s + (−4.66 + 4.66i)13-s − 0.943·15-s + (−3.43 + 2.04i)16-s + 5.33·17-s + (2.98 + 2.27i)18-s + (2.55 − 2.55i)19-s + ⋯ |
L(s) = 1 | + (0.606 − 0.795i)2-s + (−0.240 + 0.240i)3-s + (−0.264 − 0.964i)4-s + (0.506 + 0.506i)5-s + (0.0454 + 0.337i)6-s + (−0.927 − 0.374i)8-s + 0.884i·9-s + (0.709 − 0.0956i)10-s + (1.16 + 1.16i)11-s + (0.295 + 0.168i)12-s + (−1.29 + 1.29i)13-s − 0.243·15-s + (−0.859 + 0.510i)16-s + 1.29·17-s + (0.703 + 0.536i)18-s + (0.587 − 0.587i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98379 + 0.142446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98379 + 0.142446i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.857 + 1.12i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.416 - 0.416i)T - 3iT^{2} \) |
| 5 | \( 1 + (-1.13 - 1.13i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.85 - 3.85i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.66 - 4.66i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + (-2.55 + 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.60iT - 23T^{2} \) |
| 29 | \( 1 + (1.22 - 1.22i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.833T + 31T^{2} \) |
| 37 | \( 1 + (4.42 + 4.42i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.263iT - 41T^{2} \) |
| 43 | \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + (-0.0476 - 0.0476i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.60 - 3.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.46 - 4.46i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.50 + 9.50i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.05iT - 71T^{2} \) |
| 73 | \( 1 + 5.48iT - 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 + (5.84 - 5.84i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.32iT - 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−10.28888814192041243256994282221, −9.690388800634393961809262859310, −9.176127272540091177245013883572, −7.43362219433981301026746815935, −6.77290307336528790657062052091, −5.60592127040235087117428154880, −4.79471581124356899091269679357, −3.97906835495047425285243016603, −2.55271419548687756985818221407, −1.69551579924970737754873145237,
0.913032474368678230306694346627, 3.02994039812847438531109751740, 3.85623597825475476239059292228, 5.35810316779159236962620701378, 5.67246211335548104324164916950, 6.64079227959161878827448727497, 7.56539858138328038728663307094, 8.436126474504863117224350337068, 9.307289847669562065404434550166, 10.02893403392449393643311056322