L(s) = 1 | − 3-s − 7-s + 13-s − 1.41i·17-s − 1.41i·19-s + 21-s − 1.41i·23-s − 25-s + 27-s + 1.41i·31-s − 1.41i·37-s − 39-s − 41-s − 43-s − 1.41i·47-s + ⋯ |
L(s) = 1 | − 3-s − 7-s + 13-s − 1.41i·17-s − 1.41i·19-s + 21-s − 1.41i·23-s − 25-s + 27-s + 1.41i·31-s − 1.41i·37-s − 39-s − 41-s − 43-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4918201980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4918201980\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - 1.41iT - T^{2} \) |
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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
−10.06557127513212134046413779617, −9.081091639799794217505974475719, −8.448138664324486355840241025158, −6.86465006952430357066350568685, −6.72761292918847687038190548176, −5.60360923219166421438481674720, −4.91852525924342045121530134770, −3.64659253396289276181038913378, −2.57302160289005139429590833888, −0.52541443102369173707496324544,
1.56908270493525320923711799358, 3.33902535481590772358247225056, 4.05116597880959216005316560097, 5.56211887121719642792678103067, 5.99749659696074505737855818255, 6.62215177993869794287905606023, 7.900140989396057203826171889918, 8.592068185855285414828290204188, 9.881121358188980569282996482863, 10.14908862392410302643147367765