| L(s) = 1 | + 1.60·3-s + i·5-s − 4.33i·7-s − 0.439·9-s + 1.73i·11-s + 1.60i·15-s − 7.50·17-s − 5.37i·19-s − 6.93i·21-s + 1.16·23-s − 25-s − 5.50·27-s − 2.02·29-s + 7.86i·31-s + 2.77i·33-s + ⋯ |
| L(s) = 1 | + 0.923·3-s + 0.447i·5-s − 1.63i·7-s − 0.146·9-s + 0.522i·11-s + 0.413i·15-s − 1.81·17-s − 1.23i·19-s − 1.51i·21-s + 0.242·23-s − 0.200·25-s − 1.05·27-s − 0.375·29-s + 1.41i·31-s + 0.482i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6606893251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6606893251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.60T + 3T^{2} \) |
| 7 | \( 1 + 4.33iT - 7T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 17 | \( 1 + 7.50T + 17T^{2} \) |
| 19 | \( 1 + 5.37iT - 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 - 7.86iT - 31T^{2} \) |
| 37 | \( 1 + 9.53iT - 37T^{2} \) |
| 41 | \( 1 - 7.73iT - 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 6.34iT - 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 5.26iT - 67T^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 + 5.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.16T + 79T^{2} \) |
| 83 | \( 1 + 0.456iT - 83T^{2} \) |
| 89 | \( 1 + 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 2.81iT - 97T^{2} \) |
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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
−8.270109442446739073688622765892, −7.42515609795749630542388376797, −6.98348210924875816730900098728, −6.34945270081983195862146622860, −4.91501226824690967830164758025, −4.31497488149998727807518151346, −3.46743128771482986917936382382, −2.68804658863781236945279392087, −1.70166289359425860790954991561, −0.15436777573889010360149021476,
1.82120365429999100370311423133, 2.44336249274355565043184318982, 3.28954931493021210100149798914, 4.23508782546277127063209257510, 5.26021650122930482321911520255, 5.90294693352870023708188027104, 6.58632359120370621198792180910, 7.87479280570231101072792313287, 8.325635533988686544052013568448, 8.955809598764151079495629627377
