L(s) = 1 | + i·2-s − 4-s + (1.67 + 1.48i)5-s − 3.35i·7-s − i·8-s + (−1.48 + 1.67i)10-s + 0.962·11-s + 1.61i·13-s + 3.35·14-s + 16-s + 0.387i·17-s − 19-s + (−1.67 − 1.48i)20-s + 0.962i·22-s − 0.962i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.749 + 0.662i)5-s − 1.26i·7-s − 0.353i·8-s + (−0.468 + 0.529i)10-s + 0.290·11-s + 0.447i·13-s + 0.895·14-s + 0.250·16-s + 0.0940i·17-s − 0.229·19-s + (−0.374 − 0.331i)20-s + 0.205i·22-s − 0.200i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930082513\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930082513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 3.35iT - 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 - 1.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.387iT - 17T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 - 6.96T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 + 1.61iT - 37T^{2} \) |
| 41 | \( 1 - 9.27T + 41T^{2} \) |
| 43 | \( 1 + 6.18iT - 43T^{2} \) |
| 47 | \( 1 + 0.962iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 7.22iT - 67T^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 - 3.22iT - 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−9.451383012933155287369233791955, −8.597167541124059004999277480428, −7.65844727573707313166878999163, −6.89227924653223515635763654787, −6.48280300762636230151135152590, −5.53644324502021762049461089119, −4.46410380432713379481527171174, −3.73718108759838023813831609787, −2.46178742754807222267640820671, −1.00399891866353753784450633822,
1.01105291670493219602777836254, 2.21204272687266732311143998992, 2.90840349330703420590720401256, 4.26586968098719011463317028649, 5.13683674287577788071854459083, 5.80023459051981853608501175446, 6.59277484829343264116401079694, 8.048595097331937704988593062143, 8.620451228864850096933132115179, 9.294366095063741198714805488092