L(s) = 1 | − 2-s − 3-s − 4-s + 4·5-s + 6-s − 7-s + 3·8-s + 9-s − 4·10-s + 4·11-s + 12-s + 13-s + 14-s − 4·15-s − 16-s − 18-s − 4·20-s + 21-s − 4·22-s + 4·23-s − 3·24-s + 11·25-s − 26-s − 27-s + 28-s + 4·29-s + 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 1.78·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 1.03·15-s − 1/4·16-s − 0.235·18-s − 0.894·20-s + 0.218·21-s − 0.852·22-s + 0.834·23-s − 0.612·24-s + 11/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.283051350\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.283051350\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.83142560615469, −13.62071583306149, −13.08286817375556, −12.50074784227230, −12.21632927765470, −11.26237667963452, −10.85463278422808, −10.38659026472775, −9.939831679578405, −9.355138681590853, −9.182513877584764, −8.739243017017309, −8.053178032341771, −7.128503028096486, −6.822647861065819, −6.302770693274969, −5.736896368405259, −5.163356918376185, −4.834186084826077, −3.873586147537060, −3.472081172596530, −2.344106230239549, −1.852257021250577, −1.109480108097845, −0.6903223939304206,
0.6903223939304206, 1.109480108097845, 1.852257021250577, 2.344106230239549, 3.472081172596530, 3.873586147537060, 4.834186084826077, 5.163356918376185, 5.736896368405259, 6.302770693274969, 6.822647861065819, 7.128503028096486, 8.053178032341771, 8.739243017017309, 9.182513877584764, 9.355138681590853, 9.939831679578405, 10.38659026472775, 10.85463278422808, 11.26237667963452, 12.21632927765470, 12.50074784227230, 13.08286817375556, 13.62071583306149, 13.83142560615469