L(s) = 1 | + (2.02 + 0.542i)2-s − i·3-s + (2.07 + 1.19i)4-s + (1.89 − 0.507i)5-s + (0.542 − 2.02i)6-s + (−1.04 + 2.43i)7-s + (0.588 + 0.588i)8-s − 9-s + 4.10·10-s + (1.77 + 1.77i)11-s + (1.19 − 2.07i)12-s + (−3.33 − 1.36i)13-s + (−3.43 + 4.35i)14-s + (−0.507 − 1.89i)15-s + (−1.52 − 2.64i)16-s + (3.66 − 6.34i)17-s + ⋯ |
L(s) = 1 | + (1.43 + 0.383i)2-s − 0.577i·3-s + (1.03 + 0.599i)4-s + (0.846 − 0.226i)5-s + (0.221 − 0.826i)6-s + (−0.394 + 0.919i)7-s + (0.207 + 0.207i)8-s − 0.333·9-s + 1.29·10-s + (0.534 + 0.534i)11-s + (0.345 − 0.599i)12-s + (−0.925 − 0.378i)13-s + (−0.916 + 1.16i)14-s + (−0.130 − 0.488i)15-s + (−0.381 − 0.660i)16-s + (0.887 − 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.67102 + 0.118705i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.67102 + 0.118705i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (1.04 - 2.43i)T \) |
| 13 | \( 1 + (3.33 + 1.36i)T \) |
good | 2 | \( 1 + (-2.02 - 0.542i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (-1.89 + 0.507i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.77 - 1.77i)T + 11iT^{2} \) |
| 17 | \( 1 + (-3.66 + 6.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.681 - 0.681i)T + 19iT^{2} \) |
| 23 | \( 1 + (5.74 - 3.31i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.21 - 9.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.782 - 2.92i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.356 - 1.33i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.710 - 0.190i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-10.1 + 5.84i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.971 + 3.62i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.15 + 2.00i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 4.58i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 9.45iT - 61T^{2} \) |
| 67 | \( 1 + (-4.43 + 4.43i)T - 67iT^{2} \) |
| 71 | \( 1 + (-8.32 - 2.23i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-7.50 - 2.01i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.31 - 4.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.95 + 1.06i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.63 + 9.85i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−12.37811629174507555395437009961, −11.61005762597786121847002483699, −9.774822591570211886351502791307, −9.247916228288990017059952594622, −7.55102096562614096280959543959, −6.71232121372943668975516491298, −5.52416110335386467609004695450, −5.23432668550801675228764857893, −3.42152124048374019020346118999, −2.18576664647857346913799941566,
2.25934348846465236968097629069, 3.70244514693758235359881427323, 4.33747716016575458928683136720, 5.82559323411886185123594969396, 6.27194484305362689191769478135, 7.88357940624764497408084003364, 9.461729283585920785298913624360, 10.17575400500970757422871502613, 11.05023622104186973804158671216, 12.05343394262286727268661258202