L(s) = 1 | − 1.47i·2-s − 2.48i·3-s − 0.187·4-s + (0.111 + 2.23i)5-s − 3.67·6-s − 3.24i·7-s − 2.68i·8-s − 3.16·9-s + (3.30 − 0.165i)10-s − 4.18·11-s + 0.466i·12-s − 1.78i·13-s − 4.80·14-s + (5.54 − 0.278i)15-s − 4.34·16-s + 6.33i·17-s + ⋯ |
L(s) = 1 | − 1.04i·2-s − 1.43i·3-s − 0.0939·4-s + (0.0500 + 0.998i)5-s − 1.49·6-s − 1.22i·7-s − 0.947i·8-s − 1.05·9-s + (1.04 − 0.0523i)10-s − 1.26·11-s + 0.134i·12-s − 0.494i·13-s − 1.28·14-s + (1.43 − 0.0718i)15-s − 1.08·16-s + 1.53i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0500 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0500 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9803507146\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9803507146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.111 - 2.23i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.47iT - 2T^{2} \) |
| 3 | \( 1 + 2.48iT - 3T^{2} \) |
| 7 | \( 1 + 3.24iT - 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + 1.78iT - 13T^{2} \) |
| 17 | \( 1 - 6.33iT - 17T^{2} \) |
| 23 | \( 1 + 1.43iT - 23T^{2} \) |
| 29 | \( 1 - 0.339T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 2.70iT - 37T^{2} \) |
| 41 | \( 1 + 7.13T + 41T^{2} \) |
| 43 | \( 1 + 9.89iT - 43T^{2} \) |
| 47 | \( 1 + 0.445iT - 47T^{2} \) |
| 53 | \( 1 - 7.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 - 8.55iT - 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.82iT - 73T^{2} \) |
| 79 | \( 1 + 0.698T + 79T^{2} \) |
| 83 | \( 1 - 0.552iT - 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−8.476272295242539364351836692584, −7.63555281193957181839466553481, −7.20849032196935369843607664555, −6.53098613191076499279016561881, −5.68953942482013250238554800199, −4.08793474607308300479769818848, −3.26125092018134709504010754967, −2.36063805844045900802543585486, −1.58387438575969911085381352056, −0.33135595542958049181007032156,
2.17748565092820516439596629824, 3.17641447323366675744349901680, 4.64642673254686390677666004624, 5.12690342873288386485577241124, 5.48692468638862175740417510613, 6.51388965070879102058734019315, 7.69446146428180160427030076634, 8.311774403016473652421697084399, 9.164848801504380608477817521124, 9.428519873551114986187534945290