L(s) = 1 | − 6i·11-s + 6·17-s − 2i·19-s + 5·25-s + 6·41-s − 10i·43-s − 7·49-s − 6i·59-s + 14i·67-s + 2·73-s + 18i·83-s − 18·89-s + 10·97-s − 6i·107-s − 18·113-s + ⋯ |
L(s) = 1 | − 1.80i·11-s + 1.45·17-s − 0.458i·19-s + 25-s + 0.937·41-s − 1.52i·43-s − 49-s − 0.781i·59-s + 1.71i·67-s + 0.234·73-s + 1.97i·83-s − 1.90·89-s + 1.01·97-s − 0.580i·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34360 - 0.556537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34360 - 0.556537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.73399011300224094682780167024, −9.748629045392593465453810159587, −8.752249383741104104493204952603, −8.101683451899380447987433209410, −7.01262508502942731667609086082, −5.94302179554343314978777238889, −5.20468230002258523133212636366, −3.74346129075956270246346894895, −2.84090014861999626691144965850, −0.934200635766429061756925614761,
1.54436251318823123816344282010, 2.97330513125353271685642719044, 4.30249607173664221972382518148, 5.18506337235702411474690870015, 6.34988703323918439146387537483, 7.36817614489708322710826565227, 8.000257412159321582502579217237, 9.294410980318666161761720590765, 9.910766619397820435292696604536, 10.68875605215110782900760174035