| L(s) = 1 | + 3-s − 2·4-s + 5-s + 7-s + 9-s + 5·11-s − 2·12-s + 5·13-s + 15-s + 4·16-s − 17-s − 5·19-s − 2·20-s + 21-s + 23-s − 4·25-s + 27-s − 2·28-s + 6·29-s + 5·33-s + 35-s − 2·36-s − 4·37-s + 5·39-s + 7·41-s + 7·43-s − 10·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.242·17-s − 1.14·19-s − 0.447·20-s + 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.870·33-s + 0.169·35-s − 1/3·36-s − 0.657·37-s + 0.800·39-s + 1.09·41-s + 1.06·43-s − 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.083414681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.083414681\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.75621884828556, −12.05323152096152, −11.84024253844277, −11.12261879110299, −10.56526422761747, −10.42600576836003, −9.608287963246502, −9.252840884158408, −8.934523460295056, −8.607057766337573, −8.081902539742749, −7.744653592505170, −6.844769694947719, −6.570484287178175, −5.978165253564110, −5.638997991560736, −4.922538837411543, −4.275529296501990, −3.956949259173543, −3.793101755051092, −2.890685165453940, −2.347496360418880, −1.495889860877595, −1.286593102952107, −0.5450602540224040,
0.5450602540224040, 1.286593102952107, 1.495889860877595, 2.347496360418880, 2.890685165453940, 3.793101755051092, 3.956949259173543, 4.275529296501990, 4.922538837411543, 5.638997991560736, 5.978165253564110, 6.570484287178175, 6.844769694947719, 7.744653592505170, 8.081902539742749, 8.607057766337573, 8.934523460295056, 9.252840884158408, 9.608287963246502, 10.42600576836003, 10.56526422761747, 11.12261879110299, 11.84024253844277, 12.05323152096152, 12.75621884828556
