L(s) = 1 | − 5-s − 2.72i·7-s − 6.18·11-s − 1.56·13-s − 7.33·17-s + 2.41i·19-s + (−4.64 + 1.20i)23-s + 25-s − 7.31i·29-s + 5.07·31-s + 2.72i·35-s − 8.91i·37-s − 1.74i·41-s − 11.9i·43-s + 0.0866i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.03i·7-s − 1.86·11-s − 0.434·13-s − 1.78·17-s + 0.553i·19-s + (−0.967 + 0.251i)23-s + 0.200·25-s − 1.35i·29-s + 0.911·31-s + 0.461i·35-s − 1.46i·37-s − 0.272i·41-s − 1.82i·43-s + 0.0126i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3877773702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3877773702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (4.64 - 1.20i)T \) |
good | 7 | \( 1 + 2.72iT - 7T^{2} \) |
| 11 | \( 1 + 6.18T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 - 2.41iT - 19T^{2} \) |
| 29 | \( 1 + 7.31iT - 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 8.91iT - 37T^{2} \) |
| 41 | \( 1 + 1.74iT - 41T^{2} \) |
| 43 | \( 1 + 11.9iT - 43T^{2} \) |
| 47 | \( 1 - 0.0866iT - 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 + 0.102iT - 59T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 8.46iT - 67T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.00iT - 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−7.68206434113801755368893465570, −7.48144578119440717701494361701, −6.62235812199094899621905668919, −5.80114134608580297130337913946, −5.04944639344829416808041714365, −4.23694411568338225783367598300, −3.85946967661388473690077499810, −2.61091422260240475769507380437, −2.11730649473197467534620008946, −0.52951460357316900438968150089,
0.15488057489284629257156029452, 1.82749324614074127992839124587, 2.71488655112128668875660339299, 3.01289302978583258516059042669, 4.53338860670582199221678223417, 4.74643644043052519482577754422, 5.57992624826177273203077735121, 6.36367461663269429343015354175, 7.00363030430112188902698442252, 7.86669087877289457938404853316