L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.70 − 0.292i)3-s − 1.00i·4-s + (−2.12 + 0.707i)5-s + (−0.999 + 1.41i)6-s + (−1 − i)7-s + (0.707 + 0.707i)8-s + (2.82 − i)9-s + (0.999 − 2i)10-s − 1.41i·11-s + (−0.292 − 1.70i)12-s + (2 − 2i)13-s + 1.41·14-s + (−3.41 + 1.82i)15-s − 1.00·16-s + (1.41 − 1.41i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.985 − 0.169i)3-s − 0.500i·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s + (−0.377 − 0.377i)7-s + (0.250 + 0.250i)8-s + (0.942 − 0.333i)9-s + (0.316 − 0.632i)10-s − 0.426i·11-s + (−0.0845 − 0.492i)12-s + (0.554 − 0.554i)13-s + 0.377·14-s + (−0.881 + 0.472i)15-s − 0.250·16-s + (0.342 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27481 - 0.342920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27481 - 0.342920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-6 - 6i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.48 + 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (2 + 2i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-9 - 9i)T + 97iT^{2} \) |
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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
−10.23323184539171176030455292781, −9.379918970053716101588081163262, −8.393344307537646212548362917667, −7.977131460593409579295938594143, −7.03890864584422895233984119923, −6.38228307976293236413924556091, −4.80370220038786041116755867346, −3.65630461909943278392885957277, −2.77704326513832913268829726790, −0.817567009507930712467715354000,
1.48773605658242442433617504051, 2.87982686534289537442444609591, 3.81268517560746395204398509980, 4.59845458988797496856352537905, 6.27989449214908656173478994448, 7.50685730904725941946415915664, 8.075782157929036270227928590208, 8.869432139296148415266535458222, 9.544485603357143191324189534082, 10.38334829633922446549281305387