L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 3·11-s − 4·13-s − 14-s + 16-s − 2·19-s − 3·20-s − 3·22-s + 4·25-s + 4·26-s + 28-s + 6·29-s + 5·31-s − 32-s − 3·35-s − 2·37-s + 2·38-s + 3·40-s − 6·41-s + 10·43-s + 3·44-s + 6·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.670·20-s − 0.639·22-s + 4/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.507·35-s − 0.328·37-s + 0.324·38-s + 0.474·40-s − 0.937·41-s + 1.52·43-s + 0.452·44-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.51731793770337, −15.05887791208402, −14.45711489610508, −14.18036486154823, −13.27570413810657, −12.50361610202890, −12.01154234475471, −11.81059881045570, −11.26155004513572, −10.52339100895060, −10.18677451939172, −9.370297569176822, −8.873315899420236, −8.346078210322909, −7.733331211376820, −7.426786857726018, −6.691302078715823, −6.255131964865060, −5.258525174349598, −4.490901924937799, −4.188660563979319, −3.266758360267214, −2.658009140212916, −1.728618804938094, −0.8595831211664469, 0,
0.8595831211664469, 1.728618804938094, 2.658009140212916, 3.266758360267214, 4.188660563979319, 4.490901924937799, 5.258525174349598, 6.255131964865060, 6.691302078715823, 7.426786857726018, 7.733331211376820, 8.346078210322909, 8.873315899420236, 9.370297569176822, 10.18677451939172, 10.52339100895060, 11.26155004513572, 11.81059881045570, 12.01154234475471, 12.50361610202890, 13.27570413810657, 14.18036486154823, 14.45711489610508, 15.05887791208402, 15.51731793770337