| L(s) = 1 | + 3i·5-s + 1.73i·7-s − 5.19·11-s + 2·13-s − 6i·17-s + 6.92i·19-s − 4·25-s + 6i·29-s + 5.19i·31-s − 5.19·35-s − 8·37-s − 10.3i·43-s − 10.3·47-s + 4·49-s − 9i·53-s + ⋯ |
| L(s) = 1 | + 1.34i·5-s + 0.654i·7-s − 1.56·11-s + 0.554·13-s − 1.45i·17-s + 1.58i·19-s − 0.800·25-s + 1.11i·29-s + 0.933i·31-s − 0.878·35-s − 1.31·37-s − 1.58i·43-s − 1.51·47-s + 0.571·49-s − 1.23i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7298842198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7298842198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.930027860114864397234528088027, −8.850656792785135421373551271300, −8.089367699753014909268375469234, −7.26718451181649366410742070018, −6.64471895768684495816932025495, −5.58850252708836668728060585045, −5.07212918553536742522905934743, −3.47918778376496003955778359435, −2.95049089177600711613217247057, −1.93532531078738022073602960429,
0.26480066013579019922542109874, 1.50371005452550506370348559514, 2.81466538046308575267979916916, 4.11136131290627587473294741885, 4.73923558901712691720188555035, 5.55924040153964734078941035547, 6.42661248408913616280020231126, 7.61407736051520757172702792508, 8.143311529025134716401097355401, 8.826024807460950607352671764792
