L(s) = 1 | + (0.0747 − 0.997i)2-s + (−1.82 + 0.563i)3-s + (−0.988 − 0.149i)4-s + (0.425 + 1.86i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (2.19 − 1.49i)9-s + (−0.123 − 0.0841i)11-s + (1.88 − 0.284i)12-s + (−0.134 + 0.0648i)13-s + (−0.955 − 0.294i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−1.32 − 2.29i)18-s + (0.142 + 1.90i)21-s + (−0.0931 + 0.116i)22-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)2-s + (−1.82 + 0.563i)3-s + (−0.988 − 0.149i)4-s + (0.425 + 1.86i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (2.19 − 1.49i)9-s + (−0.123 − 0.0841i)11-s + (1.88 − 0.284i)12-s + (−0.134 + 0.0648i)13-s + (−0.955 − 0.294i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−1.32 − 2.29i)18-s + (0.142 + 1.90i)21-s + (−0.0931 + 0.116i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5899183442\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5899183442\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0747 + 0.997i)T \) |
| 7 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + (-0.365 + 0.930i)T \) |
good | 3 | \( 1 + (1.82 - 0.563i)T + (0.826 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 11 | \( 1 + (0.123 + 0.0841i)T + (0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.658 - 1.67i)T + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.107730744206378482766052446875, −7.62367832059738817214075124289, −7.13973759811321199765651432554, −6.03979516192960644747684883174, −5.23198014326493783744647303953, −4.89759936778790877662427857227, −3.96338913511966700290518231495, −3.32595382080973913998731687153, −1.57258342611680514737277280315, −0.64355780734717436148257273164,
0.919696082950570671571896365228, 2.33105082181338681546076180988, 4.02460923263171116028899104273, 4.91284949884844092799315555644, 5.32605200792473416004831162365, 6.13979552284261500320020296661, 6.51365563854547782335388930807, 7.25235969354013668226296304384, 8.147553521022779890387718856257, 8.676882442267301657814677411331