L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 1.41i·29-s + (−0.258 − 0.965i)32-s − 2i·43-s + (1.22 − 0.707i)44-s + (0.366 + 1.36i)46-s + (−0.707 + 0.707i)50-s + (1.22 + 0.707i)53-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + 1.41i·29-s + (−0.258 − 0.965i)32-s − 2i·43-s + (1.22 − 0.707i)44-s + (0.366 + 1.36i)46-s + (−0.707 + 0.707i)50-s + (1.22 + 0.707i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7387226811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7387226811\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.124554499606032285548011410478, −8.701477085016473379242981941355, −8.071517023645916358802118712325, −6.98284805552286638866097346565, −6.42735454756639515792821725609, −5.51529773668200935381352613389, −4.12036402573428287972363976573, −3.25031189567312950370319149255, −2.20782116348550708129413046027, −0.843969401835914170350792639920,
1.38649625387249975259853539489, 2.35958524229136551877381810066, 3.68610370810542115210532468104, 4.81354011765740171604957106611, 5.82354279982466946423044187600, 6.61781071881858760903003647943, 7.38425425483051467547392341397, 7.967898959140578750543394773075, 8.927167732103795973802557764971, 9.738960871949649149118001589949