L(s) = 1 | − 2.82i·3-s + i·5-s + 2.82·7-s − 5.00·9-s − 5.65i·11-s + 2i·13-s + 2.82·15-s + 2·17-s − 8.00i·21-s − 2.82·23-s − 25-s + 5.65i·27-s − 6i·29-s + 5.65·31-s − 16.0·33-s + ⋯ |
L(s) = 1 | − 1.63i·3-s + 0.447i·5-s + 1.06·7-s − 1.66·9-s − 1.70i·11-s + 0.554i·13-s + 0.730·15-s + 0.485·17-s − 1.74i·21-s − 0.589·23-s − 0.200·25-s + 1.08i·27-s − 1.11i·29-s + 1.01·31-s − 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.658452185\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658452185\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.034204763433559595355599062052, −8.235292219874550431507890998165, −7.81601374186909129794886578270, −6.96404900304427192139928943566, −6.08849272471504480749388027829, −5.55378169370215349425242754569, −4.05980074880724542032358770266, −2.79979616547071034496730591237, −1.84917373363336474364803851102, −0.71979849040965480667717737248,
1.65305633989897442053963579908, 3.09633989284459406934619731709, 4.22352158976485560969083953699, 4.88981382968949787566903416861, 5.21642104113449935245563080241, 6.60320709852443554873695040843, 7.907481727131496427315164156379, 8.338026772104313295440473409791, 9.482968859179606657162660724202, 9.880634095585036209324391514237