L(s) = 1 | − 2.50i·2-s + 2.26i·3-s − 4.26·4-s + (−0.605 + 2.15i)5-s + 5.66·6-s − 2.41i·7-s + 5.66i·8-s − 2.12·9-s + (5.38 + 1.51i)10-s + 0.723·11-s − 9.64i·12-s − 1.06i·13-s − 6.05·14-s + (−4.87 − 1.36i)15-s + 5.64·16-s + 6.18i·17-s + ⋯ |
L(s) = 1 | − 1.76i·2-s + 1.30i·3-s − 2.13·4-s + (−0.270 + 0.962i)5-s + 2.31·6-s − 0.913i·7-s + 2.00i·8-s − 0.706·9-s + (1.70 + 0.478i)10-s + 0.218·11-s − 2.78i·12-s − 0.294i·13-s − 1.61·14-s + (−1.25 − 0.353i)15-s + 1.41·16-s + 1.49i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477070 + 0.361422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477070 + 0.361422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.605 - 2.15i)T \) |
| 151 | \( 1 + T \) |
good | 2 | \( 1 + 2.50iT - 2T^{2} \) |
| 3 | \( 1 - 2.26iT - 3T^{2} \) |
| 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 11 | \( 1 - 0.723T + 11T^{2} \) |
| 13 | \( 1 + 1.06iT - 13T^{2} \) |
| 17 | \( 1 - 6.18iT - 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 - 7.55iT - 37T^{2} \) |
| 41 | \( 1 - 0.455T + 41T^{2} \) |
| 43 | \( 1 + 4.49iT - 43T^{2} \) |
| 47 | \( 1 - 0.268iT - 47T^{2} \) |
| 53 | \( 1 - 5.33iT - 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 9.19iT - 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 - 1.54iT - 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 0.0292T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−10.56602508013114240755241589687, −10.16583781921296151724428151573, −9.317756677412616892050971761030, −8.395940687643648560604148544466, −7.16365447770749174481003787990, −5.73777665635372734852145126953, −4.33452373496664101206846092953, −3.88877573623264722088621441606, −3.25218499310496269266614851945, −1.81267689852022301794358311742,
0.30124267363188914652765381598, 2.08292348899942520862028536158, 4.16947483889149017265925645242, 5.20900695395018940279007429671, 5.90119968964083613684745046973, 6.80196748114882400862981821393, 7.46194833816589273783363366228, 8.202278455268723126309349125797, 8.969789130382387845436617846629, 9.402760840182152891677653096162