| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 14-s − 15-s + 16-s − 18-s + 4·19-s − 20-s + 21-s − 4·22-s − 2·23-s − 24-s + 25-s + 27-s + 28-s + 8·29-s + 30-s − 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.48·29-s + 0.182·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.568755106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.568755106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.83817933217497, −14.52447493731784, −13.96757990266483, −13.55892513610165, −12.66349824968092, −12.13391112425623, −11.83790830929809, −11.12667318139042, −10.76365730231916, −9.945381687249142, −9.529147215852830, −9.021769752189554, −8.501118413753984, −7.955160276644203, −7.489214939829714, −6.882073136598036, −6.370394549233409, −5.614209956685755, −4.830846895883620, −4.105381558081510, −3.611110407670594, −2.861783526505263, −2.134461840526893, −1.310442011790466, −0.7173084537909576,
0.7173084537909576, 1.310442011790466, 2.134461840526893, 2.861783526505263, 3.611110407670594, 4.105381558081510, 4.830846895883620, 5.614209956685755, 6.370394549233409, 6.882073136598036, 7.489214939829714, 7.955160276644203, 8.501118413753984, 9.021769752189554, 9.529147215852830, 9.945381687249142, 10.76365730231916, 11.12667318139042, 11.83790830929809, 12.13391112425623, 12.66349824968092, 13.55892513610165, 13.96757990266483, 14.52447493731784, 14.83817933217497
