L(s) = 1 | + (−1.19 − 0.689i)2-s − 2.88·3-s + (−0.0491 − 0.0850i)4-s + (−0.697 + 0.402i)5-s + (3.44 + 1.98i)6-s + 2.89i·8-s + 5.30·9-s + 1.11·10-s − 5.27i·11-s + (0.141 + 0.245i)12-s + (2.36 − 2.72i)13-s + (2.01 − 1.16i)15-s + (1.89 − 3.28i)16-s + (0.280 + 0.485i)17-s + (−6.33 − 3.65i)18-s − 5.84i·19-s + ⋯ |
L(s) = 1 | + (−0.844 − 0.487i)2-s − 1.66·3-s + (−0.0245 − 0.0425i)4-s + (−0.312 + 0.180i)5-s + (1.40 + 0.811i)6-s + 1.02i·8-s + 1.76·9-s + 0.351·10-s − 1.58i·11-s + (0.0408 + 0.0707i)12-s + (0.656 − 0.754i)13-s + (0.519 − 0.299i)15-s + (0.474 − 0.821i)16-s + (0.0679 + 0.117i)17-s + (−1.49 − 0.861i)18-s − 1.34i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0447328 + 0.144152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0447328 + 0.144152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.36 + 2.72i)T \) |
good | 2 | \( 1 + (1.19 + 0.689i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + (0.697 - 0.402i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 17 | \( 1 + (-0.280 - 0.485i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 5.84iT - 19T^{2} \) |
| 23 | \( 1 + (0.802 - 1.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.14 + 1.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.01 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.07 - 0.620i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.803 - 0.463i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.32 - 1.92i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.72 - 4.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.52 - 5.49i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 7.30T + 61T^{2} \) |
| 67 | \( 1 + 7.34iT - 67T^{2} \) |
| 71 | \( 1 + (8.06 + 4.65i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.33 - 2.50i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.68 + 9.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.81iT - 83T^{2} \) |
| 89 | \( 1 + (4.33 + 2.50i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.22 + 5.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.43817994843934870094093200815, −9.363532218998560009259016612041, −8.471398001335411889194880443041, −7.48238386518554884044393973081, −6.16011569608929446637325300262, −5.69788138497467010612299407991, −4.69896195564806640700607450577, −3.15668939565791682070067643918, −1.17185208351360676201559859086, −0.16941092966093580448331921964,
1.48468076077077028196617772411, 4.01997017550761137142769346771, 4.67288849804720277198202453107, 5.99201110899995052070227356753, 6.68333481826606675631954504535, 7.48425045696469546802622495532, 8.345248230783472386001733238499, 9.599729592795075241932026326028, 10.06508672011972468859616939398, 11.00204777782873654032664962962