| L(s) = 1 | − 2i·2-s + 4i·3-s − 4·4-s + 8·6-s − 20i·7-s + 8i·8-s + 11·9-s − 48·11-s − 16i·12-s + 13i·13-s − 40·14-s + 16·16-s − 66i·17-s − 22i·18-s + 16·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.769i·3-s − 0.5·4-s + 0.544·6-s − 1.07i·7-s + 0.353i·8-s + 0.407·9-s − 1.31·11-s − 0.384i·12-s + 0.277i·13-s − 0.763·14-s + 0.250·16-s − 0.941i·17-s − 0.288i·18-s + 0.193·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4668331504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4668331504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 13iT \) |
| good | 3 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 + 20iT - 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 16T + 6.85e3T^{2} \) |
| 23 | \( 1 - 168iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 468iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 558iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 96T + 2.05e5T^{2} \) |
| 61 | \( 1 + 826T + 2.26e5T^{2} \) |
| 67 | \( 1 - 160iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 776T + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3iT - 9.12e5T^{2} \) |
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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
−10.46981037929358357764137321690, −9.772692227242943324163735988247, −9.091506929495486497099672626362, −7.72090633077006496068516612724, −7.19125039169025056540323711370, −5.53005945169771950057108686758, −4.69751010487668519691658652356, −3.86565708537775184062745872874, −2.86822852716602652965152290696, −1.33115516173750691639272353277,
0.13540187208211934873868379300, 1.80940977441005241260303504138, 2.96686608896966313020907627971, 4.57427278200835451675999541981, 5.50515954543929013853656561031, 6.34481307306828691396807476329, 7.16540188291916821623755872524, 8.288382831270347938451895239214, 8.442260192682996979237281199618, 9.884538827655131612942170749078
