L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.23 − 0.144i)5-s + (2.37 + 1.17i)7-s − 0.999·8-s + (−0.990 − 2.00i)10-s − 0.745i·11-s + (−1.67 − 2.89i)13-s + (0.168 + 2.64i)14-s + (−0.5 − 0.866i)16-s + (4.20 − 2.42i)17-s + (6.50 + 3.75i)19-s + (1.24 − 1.86i)20-s + (0.646 − 0.372i)22-s − 1.86·23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.997 − 0.0647i)5-s + (0.896 + 0.443i)7-s − 0.353·8-s + (−0.313 − 0.633i)10-s − 0.224i·11-s + (−0.463 − 0.802i)13-s + (0.0451 + 0.705i)14-s + (−0.125 − 0.216i)16-s + (1.01 − 0.588i)17-s + (1.49 + 0.861i)19-s + (0.277 − 0.415i)20-s + (0.137 − 0.0795i)22-s − 0.389·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.869700511\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.869700511\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.144i)T \) |
| 7 | \( 1 + (-2.37 - 1.17i)T \) |
good | 11 | \( 1 + 0.745iT - 11T^{2} \) |
| 13 | \( 1 + (1.67 + 2.89i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.20 + 2.42i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.50 - 3.75i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + (-0.644 - 0.371i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.33 + 2.50i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.78 - 3.33i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.849 - 1.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.64 - 4.41i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (10.0 - 5.80i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.94 - 5.09i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.33 + 4.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.48 - 0.856i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.92 - 2.26i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.17iT - 71T^{2} \) |
| 73 | \( 1 + (-5.10 - 8.84i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.10 - 5.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.26 + 1.30i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.58 + 13.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.32 - 4.01i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.259318605004643629718016152526, −8.145492712647373905214877339170, −7.82626368996450789982178333296, −7.33377111691562995350834412104, −5.99952099905635410552280772619, −5.32668073694394952272923697030, −4.64726474838550451895903707496, −3.59681372857382839200955110040, −2.80034917762782612677354848130, −1.02757236631431656971763332247,
0.812751713597151015114078809216, 2.01901467952663246589337370687, 3.29161798912784208692696808123, 4.04122294403592489004468117746, 4.81407797510097258272625144423, 5.53244592439052785263150678129, 6.90873241195011318698151355483, 7.50263495741495649082035691602, 8.206763474401362653858436695225, 9.166034351051697701811099312885