L(s) = 1 | + (0.687 − 2.56i)2-s + (0.866 − 0.5i)3-s + (−4.37 − 2.52i)4-s + (1.17 − 1.17i)5-s + (−0.687 − 2.56i)6-s + (2.40 + 1.09i)7-s + (−5.73 + 5.73i)8-s + (0.499 − 0.866i)9-s + (−2.21 − 3.83i)10-s + (−4.50 − 1.20i)11-s − 5.05·12-s + (3.59 − 0.264i)13-s + (4.46 − 5.42i)14-s + (0.431 − 1.60i)15-s + (5.71 + 9.90i)16-s + (−3.15 + 5.46i)17-s + ⋯ |
L(s) = 1 | + (0.486 − 1.81i)2-s + (0.499 − 0.288i)3-s + (−2.18 − 1.26i)4-s + (0.526 − 0.526i)5-s + (−0.280 − 1.04i)6-s + (0.910 + 0.414i)7-s + (−2.02 + 2.02i)8-s + (0.166 − 0.288i)9-s + (−0.699 − 1.21i)10-s + (−1.35 − 0.363i)11-s − 1.45·12-s + (0.997 − 0.0733i)13-s + (1.19 − 1.44i)14-s + (0.111 − 0.415i)15-s + (1.42 + 2.47i)16-s + (−0.765 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.188657 - 1.80321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.188657 - 1.80321i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.40 - 1.09i)T \) |
| 13 | \( 1 + (-3.59 + 0.264i)T \) |
good | 2 | \( 1 + (-0.687 + 2.56i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.17 + 1.17i)T - 5iT^{2} \) |
| 11 | \( 1 + (4.50 + 1.20i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (3.15 - 5.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 4.08i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.526 + 0.912i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.61 + 5.61i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.572 - 0.153i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.24 + 0.333i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (9.27 + 5.35i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.85 + 2.85i)T + 47iT^{2} \) |
| 53 | \( 1 + 0.398T + 53T^{2} \) |
| 59 | \( 1 + (-8.26 + 2.21i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (4.22 + 2.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.76 - 10.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-4.31 + 1.15i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.935 - 0.935i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.927T + 79T^{2} \) |
| 83 | \( 1 + (7.79 - 7.79i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.28 - 4.78i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.65 - 6.18i)T + (-84.0 + 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.43960018975875144438182995833, −10.71145815833064224129429521363, −9.872705706632293662670198439069, −8.620682606937609814136926783574, −8.283441911193864492619456167107, −5.87728092615200830956390235466, −5.00242826530903910989668366845, −3.76029287423391453292402368328, −2.41856281651211037914324413797, −1.43168628435736136488469163433,
2.97851569608286557496292466089, 4.63961840381731141740825048972, 5.15112600780124992438249310825, 6.57028603614772122741377749427, 7.34582236510105656914354407656, 8.225153045187732369370118871097, 9.050149879995892529113707954745, 10.21665560889663599600613048736, 11.35337284411259895565363793179, 13.09498336787262164919180400424