| L(s) = 1 | + 3·3-s − 2·4-s + 7-s + 6·9-s − 11-s − 6·12-s + 4·13-s + 4·16-s − 2·17-s − 6·19-s + 3·21-s + 5·23-s + 9·27-s − 2·28-s + 10·29-s + 31-s − 3·33-s − 12·36-s + 5·37-s + 12·39-s − 2·41-s + 8·43-s + 2·44-s − 8·47-s + 12·48-s + 49-s − 6·51-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.73·12-s + 1.10·13-s + 16-s − 0.485·17-s − 1.37·19-s + 0.654·21-s + 1.04·23-s + 1.73·27-s − 0.377·28-s + 1.85·29-s + 0.179·31-s − 0.522·33-s − 2·36-s + 0.821·37-s + 1.92·39-s − 0.312·41-s + 1.21·43-s + 0.301·44-s − 1.16·47-s + 1.73·48-s + 1/7·49-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.861971455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.861971455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.775333316288229488909333141760, −8.625648239241220640560591132688, −8.065874082970369703156109941253, −7.08385855347286904023536757123, −6.06409104376863881762815948425, −4.71497180621284015571650350496, −4.22429383069433011298913305323, −3.30387260619891210437931707303, −2.42185367175297035859813164255, −1.15748871397988376328003426573,
1.15748871397988376328003426573, 2.42185367175297035859813164255, 3.30387260619891210437931707303, 4.22429383069433011298913305323, 4.71497180621284015571650350496, 6.06409104376863881762815948425, 7.08385855347286904023536757123, 8.065874082970369703156109941253, 8.625648239241220640560591132688, 8.775333316288229488909333141760
