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
−12.01372016912809342749285172174, −11.12320549034651232018049959416, −10.24535189912948783281184977550, −9.302592474543976489563525055459, −8.611554033917077913091286940942, −7.44423218924716347098249162256, −5.76908640453884171589561568387, −5.29423475529553211433386184685, −3.80587507684635087489723777263, −2.63549969015813420091347628242,
1.51565973822216203176734940741, 3.23349964907802010162953016143, 4.69009675734075972868552749914, 6.10915812908886137955795875640, 7.05929133007431612511485815057, 7.72608082543691412317778600171, 8.873022695828254400712013405300, 9.929993887286861233702550480290, 11.11616451565845491899036330357, 11.78061059495766263985688721443