L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + 0.618·7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)12-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)19-s + (−0.500 + 0.363i)21-s + (0.309 + 0.951i)27-s + (−0.500 + 0.363i)28-s + (−0.5 − 0.363i)31-s + (0.309 + 0.951i)36-s + (−0.5 + 1.53i)37-s + (−0.5 − 1.53i)39-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + 0.618·7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)12-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)19-s + (−0.500 + 0.363i)21-s + (0.309 + 0.951i)27-s + (−0.500 + 0.363i)28-s + (−0.5 − 0.363i)31-s + (0.309 + 0.951i)36-s + (−0.5 + 1.53i)37-s + (−0.5 − 1.53i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6217266996\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6217266996\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.709801600854466782191027658194, −9.054429600531505502851408590151, −8.216489450574503323291594911023, −7.35052292903226591449958913030, −6.53022477808180324939874451517, −5.38811969545121290315492708145, −4.82597790451997536362063775927, −4.09563160253485804502416633667, −3.25263007988678201196341920083, −1.52448047513697138936080967324,
0.56230940006036325223896609102, 1.74002364490644242126589617195, 3.18011638572367583478210984492, 4.56497426577706317134459868394, 5.32144951674734200401007261215, 5.55970737874773211973068378782, 6.76861172732842154367464009487, 7.59299229130843042027984330313, 8.222792893167532898061661961407, 9.184927056966889439805008367462