L(s) = 1 | − 2.97·3-s − 4.29i·5-s + 2.64i·7-s + 5.82·9-s + 0.737i·13-s + 12.7i·15-s − 5.43·19-s − 7.86i·21-s − 7.48i·23-s − 13.4·25-s − 8.40·27-s + 11.3·35-s − 2.19i·39-s − 25.0i·45-s − 7.00·49-s + ⋯ |
L(s) = 1 | − 1.71·3-s − 1.92i·5-s + 0.999i·7-s + 1.94·9-s + 0.204i·13-s + 3.29i·15-s − 1.24·19-s − 1.71i·21-s − 1.56i·23-s − 2.69·25-s − 1.61·27-s + 1.92·35-s − 0.350i·39-s − 3.73i·45-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2263884512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2263884512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 + 4.29iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 0.737iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 + 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.419387341127903377572974709247, −8.710986346225082605532573759347, −8.125995424215652808192904258209, −6.77332003109129204271586760156, −6.00144130253829915412993337806, −5.46237635344906689031213449332, −4.68849615750409298595453193903, −4.24453835660408407407337231943, −2.13886816851499135542650551143, −0.986317556213917886030009066632,
0.12993200639391534329578068949, 1.79543162845237260553689733651, 3.28765269988287909930543923491, 4.08845778810598428272259262726, 5.13493925460982108882792738468, 6.23081803561415316837444627312, 6.44756814050234545694316060021, 7.30941011644690789015775287625, 7.78507965081744126606653517099, 9.521844503331526125844106771072