L(s) = 1 | + 2.27i·3-s − 2.55i·5-s + 1.00i·7-s − 2.18·9-s − 4.20i·11-s − 6.65·13-s + 5.82·15-s + (2.10 + 3.54i)17-s + 1.96·19-s − 2.28·21-s + 7.06i·23-s − 1.54·25-s + 1.86i·27-s − 2.50i·29-s − 7.19i·31-s + ⋯ |
L(s) = 1 | + 1.31i·3-s − 1.14i·5-s + 0.379i·7-s − 0.727·9-s − 1.26i·11-s − 1.84·13-s + 1.50·15-s + (0.511 + 0.859i)17-s + 0.450·19-s − 0.499·21-s + 1.47i·23-s − 0.309·25-s + 0.358i·27-s − 0.464i·29-s − 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.210722812\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.210722812\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-2.10 - 3.54i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 2.27iT - 3T^{2} \) |
| 5 | \( 1 + 2.55iT - 5T^{2} \) |
| 7 | \( 1 - 1.00iT - 7T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 - 7.06iT - 23T^{2} \) |
| 29 | \( 1 + 2.50iT - 29T^{2} \) |
| 31 | \( 1 + 7.19iT - 31T^{2} \) |
| 37 | \( 1 + 5.03iT - 37T^{2} \) |
| 41 | \( 1 - 11.6iT - 41T^{2} \) |
| 43 | \( 1 - 2.35T + 43T^{2} \) |
| 47 | \( 1 + 9.91T + 47T^{2} \) |
| 53 | \( 1 - 0.260T + 53T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 + 5.64iT - 71T^{2} \) |
| 73 | \( 1 - 8.82iT - 73T^{2} \) |
| 79 | \( 1 - 13.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.15T + 83T^{2} \) |
| 89 | \( 1 - 4.88T + 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−8.891579503158709359295223121914, −8.066663325408308920700299678872, −7.52735109241823452545275492140, −6.15139306735432773589336666943, −5.38478936992976335224114734848, −5.06696874356507171157901595994, −4.17229189706804644614354898448, −3.49423613674064536583679836031, −2.48054942096953225928705782070, −1.06819940268752361039243213948,
0.37951600855096401177276261124, 1.79091659840656295209464416195, 2.50597514567892720071766701694, 3.18324911116128212404143615089, 4.57900507903427989081140435097, 5.15780372064458561746372565767, 6.42231638927380243221700788839, 7.05236767634348733518875863830, 7.19890682559577117497582117170, 7.76251419057198099330443866763