| L(s) = 1 | − 3-s − 4.03·5-s − 3.90·7-s + 9-s + 4.96·11-s + 0.829·13-s + 4.03·15-s + 0.147·17-s − 2.16·19-s + 3.90·21-s − 23-s + 11.2·25-s − 27-s − 29-s − 9.11·31-s − 4.96·33-s + 15.7·35-s + 6.18·37-s − 0.829·39-s − 4.10·41-s + 1.53·43-s − 4.03·45-s − 6.98·47-s + 8.24·49-s − 0.147·51-s − 9.48·53-s − 19.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.80·5-s − 1.47·7-s + 0.333·9-s + 1.49·11-s + 0.230·13-s + 1.04·15-s + 0.0356·17-s − 0.497·19-s + 0.852·21-s − 0.208·23-s + 2.24·25-s − 0.192·27-s − 0.185·29-s − 1.63·31-s − 0.863·33-s + 2.66·35-s + 1.01·37-s − 0.132·39-s − 0.641·41-s + 0.234·43-s − 0.600·45-s − 1.01·47-s + 1.17·49-s − 0.0205·51-s − 1.30·53-s − 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3765069071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3765069071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 4.03T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 - 0.829T + 13T^{2} \) |
| 17 | \( 1 - 0.147T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 31 | \( 1 + 9.11T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 + 4.01T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.59973079241012091959435338021, −7.18206642010979568913560030532, −6.34997288830916147297600521852, −6.13022260890342075826206095520, −4.83136007270699392474862465115, −4.13741182659482740269506888241, −3.60963226559826214650133382526, −3.08389133805333103192864591277, −1.50408474063427632319639874927, −0.32269130001284602733233821822,
0.32269130001284602733233821822, 1.50408474063427632319639874927, 3.08389133805333103192864591277, 3.60963226559826214650133382526, 4.13741182659482740269506888241, 4.83136007270699392474862465115, 6.13022260890342075826206095520, 6.34997288830916147297600521852, 7.18206642010979568913560030532, 7.59973079241012091959435338021
