L(s) = 1 | + (0.951 + 0.309i)3-s + (−1.64 − 1.51i)5-s − 3.78i·7-s + (0.809 + 0.587i)9-s + (0.653 − 0.474i)11-s + (2.79 − 3.84i)13-s + (−1.09 − 1.95i)15-s + (−1.09 + 0.355i)17-s + (−0.00463 − 0.0142i)19-s + (1.17 − 3.60i)21-s + (3.68 + 5.07i)23-s + (0.395 + 4.98i)25-s + (0.587 + 0.809i)27-s + (−1.14 + 3.51i)29-s + (−0.488 − 1.50i)31-s + ⋯ |
L(s) = 1 | + (0.549 + 0.178i)3-s + (−0.734 − 0.678i)5-s − 1.43i·7-s + (0.269 + 0.195i)9-s + (0.197 − 0.143i)11-s + (0.774 − 1.06i)13-s + (−0.282 − 0.503i)15-s + (−0.264 + 0.0861i)17-s + (−0.00106 − 0.00327i)19-s + (0.255 − 0.786i)21-s + (0.768 + 1.05i)23-s + (0.0790 + 0.996i)25-s + (0.113 + 0.155i)27-s + (−0.212 + 0.653i)29-s + (−0.0878 − 0.270i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18074 - 0.685335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18074 - 0.685335i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (1.64 + 1.51i)T \) |
good | 7 | \( 1 + 3.78iT - 7T^{2} \) |
| 11 | \( 1 + (-0.653 + 0.474i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.79 + 3.84i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.355i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.00463 + 0.0142i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.68 - 5.07i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.14 - 3.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.488 + 1.50i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.02 + 6.91i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.30 + 6.75i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (-0.500 - 0.162i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.80 + 0.911i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.25 - 6.72i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.54 - 1.84i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-12.6 + 4.10i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.51 - 4.67i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.75 - 3.78i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.86 - 8.81i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.35 - 0.439i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (13.0 - 9.46i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (7.66 + 2.49i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.36923830959154735620382933018, −10.74677004766860986591196144363, −9.662494991010611998914483710210, −8.623886910564537810170765832651, −7.81122193984947823031898854639, −7.01682807317215518278052331771, −5.37420454977508109414860600433, −4.11645782578869651777328255987, −3.40596476388928154901559020967, −1.05759334084503902380715569559,
2.18067387694999681322047917812, 3.33810073764607272107565668718, 4.63727425366019339264523641583, 6.23310047461744925884079387481, 6.96964064014825991802955267986, 8.332626704258982482632946099101, 8.806156029299838631662656298877, 9.916490971260867746718392440055, 11.24803074334097971853391350671, 11.78441950507315464546922438544