| L(s) = 1 | + 5.19·7-s − 18.9i·11-s + 13i·13-s + 32.8i·17-s − 15.5i·19-s + 18.9·23-s − 32.8·29-s + 32.9i·31-s + 46i·37-s − 32.8·41-s − 12.1·43-s − 37.9·47-s − 22·49-s − 32.8i·53-s − 37.9i·59-s + ⋯ |
| L(s) = 1 | + 0.742·7-s − 1.72i·11-s + i·13-s + 1.93i·17-s − 0.820i·19-s + 0.824·23-s − 1.13·29-s + 1.06i·31-s + 1.24i·37-s − 0.801·41-s − 0.281·43-s − 0.807·47-s − 0.448·49-s − 0.620i·53-s − 0.643i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.573766107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573766107\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 5.19T + 49T^{2} \) |
| 11 | \( 1 + 18.9iT - 121T^{2} \) |
| 13 | \( 1 - 13iT - 169T^{2} \) |
| 17 | \( 1 - 32.8iT - 289T^{2} \) |
| 19 | \( 1 + 15.5iT - 361T^{2} \) |
| 23 | \( 1 - 18.9T + 529T^{2} \) |
| 29 | \( 1 + 32.8T + 841T^{2} \) |
| 31 | \( 1 - 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 46iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 12.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 26iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 131.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 23iT - 9.40e3T^{2} \) |
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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
−8.372909993465692557980158158851, −8.224672668206333778620682468761, −6.91844187442301769425794609063, −6.43846294908967114007656233609, −5.52301822061627494586870116053, −4.85062412390027749530707109716, −3.84227542996042204045523502164, −3.19545831349094453933364390932, −1.92308334189609249875723493696, −1.11685063501231013649786412775,
0.34583162179721139647557599187, 1.63332588958570579167419589063, 2.44446936282339729122366230506, 3.47430059119731415182573536752, 4.55289398130286760434483239044, 5.05897918883530176558347800052, 5.77543240545880714446463973558, 6.98127840876192737827834551191, 7.46953573760193362323757007894, 7.964406207315710650753503778933
