L(s) = 1 | + (1 + i)2-s + (−2.74 + 1.21i)3-s + 2i·4-s + (−3.32 + 3.73i)5-s + (−3.95 − 1.52i)6-s + (−6.61 − 2.29i)7-s + (−2 + 2i)8-s + (6.03 − 6.67i)9-s + (−7.05 + 0.417i)10-s − 10.5i·11-s + (−2.43 − 5.48i)12-s + (14.9 − 14.9i)13-s + (−4.31 − 8.90i)14-s + (4.55 − 14.2i)15-s − 4·16-s + (−15.4 + 15.4i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.913 + 0.405i)3-s + 0.5i·4-s + (−0.664 + 0.747i)5-s + (−0.659 − 0.254i)6-s + (−0.944 − 0.327i)7-s + (−0.250 + 0.250i)8-s + (0.670 − 0.741i)9-s + (−0.705 + 0.0417i)10-s − 0.961i·11-s + (−0.202 − 0.456i)12-s + (1.15 − 1.15i)13-s + (−0.308 − 0.636i)14-s + (0.303 − 0.952i)15-s − 0.250·16-s + (−0.911 + 0.911i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00209134 - 0.00319202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00209134 - 0.00319202i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (2.74 - 1.21i)T \) |
| 5 | \( 1 + (3.32 - 3.73i)T \) |
| 7 | \( 1 + (6.61 + 2.29i)T \) |
good | 11 | \( 1 + 10.5iT - 121T^{2} \) |
| 13 | \( 1 + (-14.9 + 14.9i)T - 169iT^{2} \) |
| 17 | \( 1 + (15.4 - 15.4i)T - 289iT^{2} \) |
| 19 | \( 1 + 17.3T + 361T^{2} \) |
| 23 | \( 1 + (23.1 - 23.1i)T - 529iT^{2} \) |
| 29 | \( 1 + 23.7T + 841T^{2} \) |
| 31 | \( 1 + 33.1iT - 961T^{2} \) |
| 37 | \( 1 + (17.6 - 17.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 11.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (22.8 + 22.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (12.6 - 12.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (15.3 - 15.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 31.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (77.5 - 77.5i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 60.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-3.52 + 3.52i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 99.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (16.9 + 16.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 17.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (34.7 + 34.7i)T + 9.40e3iT^{2} \) |
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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
−12.96045437497089105683182041756, −11.74248398763484621355459156970, −10.90152341007884545695862293468, −10.26937359275893131396606368396, −8.705162275065337444991556250954, −7.53031402539977719587685604308, −6.21988158015379442020648249709, −5.96778499246671826629331500657, −4.06124124764045668082540632331, −3.42594967993047774969101462213,
0.00196158479429059965389806376, 1.91287279980020510431881795961, 4.00244690846778345296332605105, 4.85787818316516296374409701729, 6.25122040807103094964304353071, 6.98912266119565882417191255011, 8.622520815391639221385267281573, 9.632333559190517639998023560017, 10.86795886983471276332250162863, 11.67441522198417461002194724994