| L(s) = 1 | − 2·5-s − 7-s − 4·11-s − 6·13-s − 2·17-s + 4·19-s − 8·23-s − 25-s − 2·29-s + 2·35-s + 10·37-s + 6·41-s + 4·43-s + 49-s + 6·53-s + 8·55-s + 4·59-s − 6·61-s + 12·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s − 4·83-s + 4·85-s + 6·89-s + 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 0.338·35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.520·59-s − 0.768·61-s + 1.48·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 0.439·83-s + 0.433·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6438021534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6438021534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.198359639724786266519663657681, −7.53820374567136060820384434121, −7.40972678115170708004650339469, −6.17138939899101715746472197303, −5.44976155260262292224035849332, −4.58738843418466037711840623962, −3.94021406301247359498429398492, −2.84972962383490204420977296978, −2.22731284276927216248003840864, −0.42765725228175955742967455660,
0.42765725228175955742967455660, 2.22731284276927216248003840864, 2.84972962383490204420977296978, 3.94021406301247359498429398492, 4.58738843418466037711840623962, 5.44976155260262292224035849332, 6.17138939899101715746472197303, 7.40972678115170708004650339469, 7.53820374567136060820384434121, 8.198359639724786266519663657681
