| L(s) = 1 | − 2.82i·5-s − 2.82·7-s − 4i·11-s + 5.65i·13-s + 2·17-s − 4i·19-s − 5.65·23-s − 3.00·25-s − 2.82i·29-s − 8.48·31-s + 8.00i·35-s − 10·41-s + 12i·43-s − 5.65·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s − 1.06·7-s − 1.20i·11-s + 1.56i·13-s + 0.485·17-s − 0.917i·19-s − 1.17·23-s − 0.600·25-s − 0.525i·29-s − 1.52·31-s + 1.35i·35-s − 1.56·41-s + 1.82i·43-s − 0.825·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4773415956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4773415956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.307471844082756920649616638749, −8.750156231695827763474842223201, −7.910520567878033664996231257736, −6.70989442004737162871460805361, −6.06172355475996944645834051201, −5.07270764464667137447942657952, −4.12678273599902205904955052875, −3.18376368063258798649911542502, −1.66601380241701012587719386530, −0.19731264246088951185321303071,
2.02931349115505972846574068160, 3.20577265583575093578349322230, 3.71908674026926067478169834057, 5.29983294506024541734843904276, 6.07217901468720812716646738744, 7.00150158332831843107301464615, 7.49556416897967495425423867147, 8.498400794445151053345552947257, 9.829906510702137592133832334028, 10.11520672881253264555485690056
