L(s) = 1 | + (−0.569 + 1.29i)2-s + (−1.35 − 1.47i)4-s − 0.772i·5-s − 7-s + (2.67 − 0.908i)8-s + (1 + 0.440i)10-s + 4.40i·11-s − 5.89i·13-s + (0.569 − 1.29i)14-s + (−0.350 + 3.98i)16-s + 4.21·17-s − 5.01i·19-s + (−1.13 + 1.04i)20-s + (−5.70 − 2.50i)22-s + 8.77·23-s + ⋯ |
L(s) = 1 | + (−0.402 + 0.915i)2-s + (−0.675 − 0.737i)4-s − 0.345i·5-s − 0.377·7-s + (0.947 − 0.321i)8-s + (0.316 + 0.139i)10-s + 1.32i·11-s − 1.63i·13-s + (0.152 − 0.345i)14-s + (−0.0876 + 0.996i)16-s + 1.02·17-s − 1.15i·19-s + (−0.254 + 0.233i)20-s + (−1.21 − 0.535i)22-s + 1.82·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05483 + 0.173934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05483 + 0.173934i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.569 - 1.29i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 0.772iT - 5T^{2} \) |
| 11 | \( 1 - 4.40iT - 11T^{2} \) |
| 13 | \( 1 + 5.89iT - 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 + 5.01iT - 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 + 3.63iT - 29T^{2} \) |
| 31 | \( 1 - 7.40T + 31T^{2} \) |
| 37 | \( 1 - 5.01iT - 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 + 0.880iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6.72iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 5.89iT - 61T^{2} \) |
| 67 | \( 1 + 0.880iT - 67T^{2} \) |
| 71 | \( 1 + 4.21T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.54iT - 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−10.51507952584987395518939186356, −9.999190142673661559548412441683, −9.101387032191566374167970290753, −8.202800753413010619257095255254, −7.32443710734900379526653080308, −6.56276231112118442490375284614, −5.27015362040381311688504260823, −4.73332308725475268925012887396, −3.01218351171080009070782051668, −0.922568715449747197900600226496,
1.26031584173127557723998357608, 2.91048722861665329930037297635, 3.68240926516034957401925997177, 5.00720872725368538376112579821, 6.35464204147607771092255943731, 7.35998393445539924049029264592, 8.542119937164810181549597837463, 9.111492046378453251985740606176, 10.11949668587622685930120706975, 10.86706312864014580614465125928