| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s − 19-s − 4·23-s − 27-s − 3·29-s − 4·31-s + 4·33-s − 5·37-s + 39-s + 9·41-s − 2·43-s + 3·47-s − 7·49-s + 53-s + 57-s − 10·59-s + 4·61-s + 9·67-s + 4·69-s − 7·71-s + 4·73-s − 11·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.229·19-s − 0.834·23-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.696·33-s − 0.821·37-s + 0.160·39-s + 1.40·41-s − 0.304·43-s + 0.437·47-s − 49-s + 0.137·53-s + 0.132·57-s − 1.30·59-s + 0.512·61-s + 1.09·67-s + 0.481·69-s − 0.830·71-s + 0.468·73-s − 1.23·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7429468749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7429468749\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.94267089499459, −15.68462117561048, −14.86579563404139, −14.44190958854265, −13.65109636049686, −13.20491836772519, −12.47599858839177, −12.31207621508664, −11.34807935053154, −10.99044850322456, −10.39337587184664, −9.851056852758211, −9.265185190653422, −8.448837039520178, −7.849486102118288, −7.330303146957692, −6.667675878062800, −5.824981530402252, −5.482846027746605, −4.732518602984540, −4.085165716943247, −3.223221808886221, −2.382482751373500, −1.636704365690580, −0.3742237447158758,
0.3742237447158758, 1.636704365690580, 2.382482751373500, 3.223221808886221, 4.085165716943247, 4.732518602984540, 5.482846027746605, 5.824981530402252, 6.667675878062800, 7.330303146957692, 7.849486102118288, 8.448837039520178, 9.265185190653422, 9.851056852758211, 10.39337587184664, 10.99044850322456, 11.34807935053154, 12.31207621508664, 12.47599858839177, 13.20491836772519, 13.65109636049686, 14.44190958854265, 14.86579563404139, 15.68462117561048, 15.94267089499459
