L(s) = 1 | + (1.36 + 0.366i)2-s + 1.73i·3-s + (1.73 + i)4-s + (−0.633 + 2.36i)6-s + (2 − 1.73i)7-s + (1.99 + 2i)8-s − 0.267i·11-s + (−1.73 + 2.99i)12-s − 0.464·13-s + (3.36 − 1.63i)14-s + (1.99 + 3.46i)16-s + 6.46·17-s − 6·19-s + (2.99 + 3.46i)21-s + (0.0980 − 0.366i)22-s − 1.46·23-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + 0.999i·3-s + (0.866 + 0.5i)4-s + (−0.258 + 0.965i)6-s + (0.755 − 0.654i)7-s + (0.707 + 0.707i)8-s − 0.0807i·11-s + (−0.499 + 0.866i)12-s − 0.128·13-s + (0.899 − 0.436i)14-s + (0.499 + 0.866i)16-s + 1.56·17-s − 1.37·19-s + (0.654 + 0.755i)21-s + (0.0209 − 0.0780i)22-s − 0.305·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.41610 + 1.83154i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.41610 + 1.83154i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 11 | \( 1 + 0.267iT - 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 9.46iT - 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 1.73iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 9.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 7.46iT - 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−10.73337627893724764295264733824, −10.05312714373973623059643312231, −8.891017088999886407697848594453, −7.76555489574823451527068545417, −7.17835730567238871697323978817, −5.81167879997851008378160833835, −5.09337940252152245630653265129, −4.09079785462358892302223593350, −3.60080138354343885813528493040, −1.90623059456436271549450019071,
1.46790204204344805951313824957, 2.28883845418861232094950655946, 3.66872300217759907557538966672, 4.87725290516887990004804222771, 5.75360519527220709299336193616, 6.58772497602824339999792469557, 7.56968409200590920535286813631, 8.185059884778158665034629107433, 9.560283184392975149477488865056, 10.52254989951821015103953053337