L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.292i)22-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 − 0.292i)41-s + (1 + i)43-s + (0.707 − 0.292i)44-s + (0.707 − 0.707i)49-s − 1.00·50-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)11-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.292i)22-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 1.00i·34-s + (0.707 − 0.292i)41-s + (1 + i)43-s + (0.707 − 0.292i)44-s + (0.707 − 0.707i)49-s − 1.00·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7764748568\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7764748568\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1 - i)T + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (1 - i)T - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
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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
−9.791875644756879987676109144742, −9.241048436707080479942676643124, −8.416967790788477120431330643484, −7.42918316049116237961349697330, −7.02395623210018768690719480898, −5.93821920624968620100425069385, −5.16005311581124716762166076523, −4.18594657185239371811408309849, −2.65711377673224757773546319676, −1.26150776971721514249157610485,
1.09538257302704449829476336080, 2.46738072629543346043103425127, 3.48628519714549319517541982772, 4.34860258967616613184573277227, 5.66618825104327680111042148325, 6.64141875690812821166773048054, 7.60762134912791421563127979829, 8.366050349721793530214927732660, 9.011715341000222094396673510382, 9.848518975973238973584302436923