L(s) = 1 | + (−1.73 − 1.73i)5-s − 3.86i·7-s + (−3.34 − 3.34i)11-s + (0.267 − 0.267i)13-s − 3.46·17-s + (−3.34 + 3.34i)19-s − 1.79i·23-s + 0.999i·25-s + (−1.73 + 1.73i)29-s + 5.65·31-s + (−6.69 + 6.69i)35-s + (−3.73 − 3.73i)37-s + 6.92i·41-s + (−1.55 − 1.55i)43-s + 9.79·47-s + ⋯ |
L(s) = 1 | + (−0.774 − 0.774i)5-s − 1.46i·7-s + (−1.00 − 1.00i)11-s + (0.0743 − 0.0743i)13-s − 0.840·17-s + (−0.767 + 0.767i)19-s − 0.373i·23-s + 0.199i·25-s + (−0.321 + 0.321i)29-s + 1.01·31-s + (−1.13 + 1.13i)35-s + (−0.613 − 0.613i)37-s + 1.08i·41-s + (−0.236 − 0.236i)43-s + 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3032842172\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3032842172\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.73 + 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 + 3.86iT - 7T^{2} \) |
| 11 | \( 1 + (3.34 + 3.34i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.267 + 0.267i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + (3.34 - 3.34i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.79iT - 23T^{2} \) |
| 29 | \( 1 + (1.73 - 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (3.73 + 3.73i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (1.55 + 1.55i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.13 - 5.13i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.73 - 3.73i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4.38 + 4.38i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.79iT - 71T^{2} \) |
| 73 | \( 1 - 2.53iT - 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + (1.55 - 1.55i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−8.354930885410206418368779110991, −7.893580112860044050702430703704, −7.09857917782483819517172310610, −6.19752264643666162748143449263, −5.20829569567687368035732202732, −4.26644176446587881714769141134, −3.86949080304385174424019037631, −2.63883551679512468454216762481, −1.06060129249704437954378806888, −0.11613622188426676094541629561,
2.19880252428659570098435836258, 2.61664744286989344448470365268, 3.79532027471901828517274854163, 4.79663317650064209490453101799, 5.50432768507683331687473967835, 6.56183842909630395377065121370, 7.12944932056165091762532323314, 8.009969671791233778546121492096, 8.672570023190904352756250030623, 9.404277344580605156758241692978