L(s) = 1 | + i·3-s − 0.467i·5-s − 9-s + 4.87i·11-s − 4.56i·13-s + 0.467·15-s − 6.09·17-s − 1.34i·19-s + 4.09·23-s + 4.78·25-s − i·27-s − 7.78i·29-s + 4.40·31-s − 4.87·33-s + 4.40i·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.208i·5-s − 0.333·9-s + 1.47i·11-s − 1.26i·13-s + 0.120·15-s − 1.47·17-s − 0.308i·19-s + 0.854·23-s + 0.956·25-s − 0.192i·27-s − 1.44i·29-s + 0.791·31-s − 0.848·33-s + 0.724i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.740804830\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.740804830\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.467iT - 5T^{2} \) |
| 11 | \( 1 - 4.87iT - 11T^{2} \) |
| 13 | \( 1 + 4.56iT - 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.34iT - 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 7.78iT - 29T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 - 4.40iT - 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 4.15iT - 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 - 1.34iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 5.49iT - 61T^{2} \) |
| 67 | \( 1 - 5.90iT - 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.311519473265227161567299792454, −7.80880727072341392288401938521, −6.81530630722078169296161876395, −6.31129901447141243339773041705, −5.08654015825889404812367440192, −4.79484856171910430230601428746, −4.01507211899924195706776427793, −2.89779650486241613880755539608, −2.21898653744495384161815411832, −0.76871613858671261512015828079,
0.68860938516456647866635594606, 1.81113247460690158075582394262, 2.75935561237376134786042060455, 3.55577008909338332319608484146, 4.53539040877876207481136049977, 5.33526086167043236051249331442, 6.34794722757684943344603285353, 6.65509918536547580130561983997, 7.37534061878601075682762908493, 8.355364068682968873040555844000