L(s) = 1 | − i·3-s − i·7-s − 9-s + i·13-s + 19-s − 21-s + i·27-s − 31-s + 2i·37-s + 39-s + i·43-s − i·57-s − 61-s + i·63-s − i·67-s + ⋯ |
L(s) = 1 | − i·3-s − i·7-s − 9-s + i·13-s + 19-s − 21-s + i·27-s − 31-s + 2i·37-s + 39-s + i·43-s − i·57-s − 61-s + i·63-s − i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7589863419\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7589863419\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - 2iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−11.78061059495766263985688721443, −11.11616451565845491899036330357, −9.929993887286861233702550480290, −8.873022695828254400712013405300, −7.72608082543691412317778600171, −7.05929133007431612511485815057, −6.10915812908886137955795875640, −4.69009675734075972868552749914, −3.23349964907802010162953016143, −1.51565973822216203176734940741,
2.63549969015813420091347628242, 3.80587507684635087489723777263, 5.29423475529553211433386184685, 5.76908640453884171589561568387, 7.44423218924716347098249162256, 8.611554033917077913091286940942, 9.302592474543976489563525055459, 10.24535189912948783281184977550, 11.12320549034651232018049959416, 12.01372016912809342749285172174