L(s) = 1 | + (0.331 − 1.37i)2-s + (−1.78 − 0.910i)4-s + 2.56·5-s + 4.15i·7-s + (−1.84 + 2.14i)8-s + (0.848 − 3.52i)10-s + 2.33i·11-s + 4.29i·13-s + (5.70 + 1.37i)14-s + (2.34 + 3.24i)16-s + 17-s + (−3.68 + 2.33i)19-s + (−4.56 − 2.33i)20-s + (3.20 + 0.772i)22-s + 1.82i·23-s + ⋯ |
L(s) = 1 | + (0.234 − 0.972i)2-s + (−0.890 − 0.455i)4-s + 1.14·5-s + 1.56i·7-s + (−0.650 + 0.759i)8-s + (0.268 − 1.11i)10-s + 0.703i·11-s + 1.19i·13-s + (1.52 + 0.367i)14-s + (0.585 + 0.810i)16-s + 0.242·17-s + (−0.844 + 0.535i)19-s + (−1.01 − 0.521i)20-s + (0.683 + 0.164i)22-s + 0.379i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68881 + 0.0776714i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68881 + 0.0776714i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.331 + 1.37i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.68 - 2.33i)T \) |
good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 - 4.29iT - 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 - 1.82iT - 23T^{2} \) |
| 29 | \( 1 + 1.20iT - 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 + 5.49iT - 41T^{2} \) |
| 43 | \( 1 - 1.30iT - 43T^{2} \) |
| 47 | \( 1 + 6.99iT - 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.290T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 0.123T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 14.0iT - 89T^{2} \) |
| 97 | \( 1 + 2.41iT - 97T^{2} \) |
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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
−10.38797117034419245304813701524, −9.622081858137582721421938984266, −9.137632654494306721470630224656, −8.324929161678563675055761842540, −6.63526934390877766965823850315, −5.74667282362490633951403017540, −5.09185018788886160038224240432, −3.84112056936405333071440437844, −2.23165637808698500411772179211, −1.98796605720787608232950994818,
0.858966804449769877324218350983, 2.98778649789839206948201957157, 4.15144520768335069213123211589, 5.17556669972220479852971199406, 6.10044761173196810927530320699, 6.79191029382407345795448412717, 7.78821255975451328494256191444, 8.521060084032454733701624557978, 9.655735375719368221503257634638, 10.27775779841307283430198437689