L(s) = 1 | + (1.45 + 0.389i)2-s + (−0.866 − 1.5i)3-s + (0.232 + 0.133i)4-s + (2.90 − 2.90i)5-s + (−0.675 − 2.51i)6-s + (−1.84 − 1.84i)8-s + (−1.5 + 2.59i)9-s + (5.36 − 3.09i)10-s + (−0.285 + 1.06i)11-s − 0.464i·12-s + (−6.88 − 1.84i)15-s + (−2.23 − 3.86i)16-s + (−3.19 + 3.19i)18-s + (1.06 − 0.285i)20-s + (−0.830 + 1.43i)22-s + ⋯ |
L(s) = 1 | + (1.02 + 0.275i)2-s + (−0.499 − 0.866i)3-s + (0.116 + 0.0669i)4-s + (1.30 − 1.30i)5-s + (−0.275 − 1.02i)6-s + (−0.652 − 0.652i)8-s + (−0.5 + 0.866i)9-s + (1.69 − 0.979i)10-s + (−0.0860 + 0.321i)11-s − 0.133i·12-s + (−1.77 − 0.476i)15-s + (−0.558 − 0.966i)16-s + (−0.752 + 0.752i)18-s + (0.238 − 0.0638i)20-s + (−0.176 + 0.306i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0257 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48100 - 1.51969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48100 - 1.51969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 1.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.45 - 0.389i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (-2.90 + 2.90i)T - 5iT^{2} \) |
| 7 | \( 1 + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.285 - 1.06i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31iT^{2} \) |
| 37 | \( 1 + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-9.79 - 2.62i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.59 - 6.59i)T + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (14.8 - 3.97i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.92 + 12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (1.27 + 4.75i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + (9.29 - 9.29i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.75 - 17.7i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (84.0 - 48.5i)T^{2} \) |
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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
−10.81173966362980302365185598177, −9.640782064263691126445850641507, −9.004890051460996967445554082674, −7.82033564557144458729220294537, −6.54503631502113999127384956373, −5.84998381766898360600791289396, −5.20080855636045371637563059625, −4.38246814417981268494422555232, −2.42853375116889831060229058189, −1.03288671532338169759142507312,
2.46243456803328184106975842712, 3.33376386521510354470825606681, 4.41399924860492681751392714137, 5.72347183999337360156211713759, 5.88967740303572206829251448597, 7.11359361927102562440896111388, 8.808666967523502009295052632530, 9.590715751614117026633581052956, 10.53005391277418280781387535123, 11.01355032804935213648793694666