L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.528 + 2.17i)5-s + (0.904 − 0.904i)7-s + (0.707 − 0.707i)8-s + (1.90 − 1.16i)10-s − 2.66·11-s + (0.143 − 0.143i)13-s − 1.27·14-s − 1.00·16-s + (−3.29 + 3.29i)17-s + (4.05 + 1.59i)19-s + (−2.17 − 0.528i)20-s + (1.88 + 1.88i)22-s + (−1.75 − 1.75i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.236 + 0.971i)5-s + (0.341 − 0.341i)7-s + (0.250 − 0.250i)8-s + (0.603 − 0.367i)10-s − 0.802·11-s + (0.0398 − 0.0398i)13-s − 0.341·14-s − 0.250·16-s + (−0.799 + 0.799i)17-s + (0.930 + 0.365i)19-s + (−0.485 − 0.118i)20-s + (0.401 + 0.401i)22-s + (−0.366 − 0.366i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2209606740\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2209606740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.528 - 2.17i)T \) |
| 19 | \( 1 + (-4.05 - 1.59i)T \) |
good | 7 | \( 1 + (-0.904 + 0.904i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 + (-0.143 + 0.143i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.29 - 3.29i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.75 + 1.75i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 2.62iT - 31T^{2} \) |
| 37 | \( 1 + (0.984 + 0.984i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.74iT - 41T^{2} \) |
| 43 | \( 1 + (2.01 + 2.01i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.24 - 3.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.54 + 2.54i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 + (4.50 + 4.50i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.55iT - 71T^{2} \) |
| 73 | \( 1 + (5.25 + 5.25i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 + (6.55 + 6.55i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 + (11.8 + 11.8i)T + 97iT^{2} \) |
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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
−9.869376837950804811670819933429, −8.908266648616409164128100974914, −8.006918044476985881714956891741, −7.52272808045669960616163844852, −6.67107222149171733352275444124, −5.71589423974946576966079799421, −4.52029716946807341508325930919, −3.60025868880284714805645855869, −2.72600949928548193620377097488, −1.67178319337808354709593806692,
0.097557863576593865631639030563, 1.49575940487020992931202238431, 2.74312304024189465536487688586, 4.19878276460670013321341822057, 5.11545738640991434013476546695, 5.54370430363952847260531345694, 6.71862425133353727922430908254, 7.63761958556276347811053545622, 8.132523200434314653602231249646, 8.991969583521640085107443790909