| L(s) = 1 | − 3-s − 5-s + 5·7-s + 9-s + 5·11-s − 13-s + 15-s − 5·17-s + 3·19-s − 5·21-s − 3·23-s + 25-s − 27-s + 6·29-s + 6·31-s − 5·33-s − 5·35-s − 37-s + 39-s − 4·43-s − 45-s + 18·49-s + 5·51-s − 3·53-s − 5·55-s − 3·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s + 0.688·19-s − 1.09·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s − 0.870·33-s − 0.845·35-s − 0.164·37-s + 0.160·39-s − 0.609·43-s − 0.149·45-s + 18/7·49-s + 0.700·51-s − 0.412·53-s − 0.674·55-s − 0.397·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.284956982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284956982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.79755281325754956939176712278, −6.94284669402046062574614447628, −6.54863637384031924397796165860, −5.57664560509782476852794264609, −4.82771237006760392768657001679, −4.40130128836636760952789565531, −3.76985094489246025442451162175, −2.44390783381750474838813046979, −1.57110919670775364089695941872, −0.826200447952157064069728946270,
0.826200447952157064069728946270, 1.57110919670775364089695941872, 2.44390783381750474838813046979, 3.76985094489246025442451162175, 4.40130128836636760952789565531, 4.82771237006760392768657001679, 5.57664560509782476852794264609, 6.54863637384031924397796165860, 6.94284669402046062574614447628, 7.79755281325754956939176712278
