| L(s) = 1 | + 3-s − 2·9-s − 11-s + 12·17-s − 25-s − 5·27-s + 10·31-s − 33-s − 2·37-s − 10·49-s + 12·51-s − 2·67-s − 75-s + 81-s + 12·83-s + 10·93-s − 14·97-s + 2·99-s + 36·101-s + 16·103-s + 12·107-s − 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s + 2.91·17-s − 1/5·25-s − 0.962·27-s + 1.79·31-s − 0.174·33-s − 0.328·37-s − 1.42·49-s + 1.68·51-s − 0.244·67-s − 0.115·75-s + 1/9·81-s + 1.31·83-s + 1.03·93-s − 1.42·97-s + 0.201·99-s + 3.58·101-s + 1.57·103-s + 1.16·107-s − 0.189·111-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028736108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028736108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.041524989863110448096712863922, −8.585866889119459033715626592235, −8.132665807481574488057539990847, −7.63173684139913449353926662917, −7.60852684015633966980596452192, −6.66019819116288679958563294672, −6.08718240220327616520154502572, −5.73530160989426067882261609281, −5.11227132195889762323685903220, −4.68022649879253263009993418914, −3.70217639365433698551581447714, −3.27381985666804460223319703873, −2.86631612844463412939210052570, −1.95567073598448482222279023681, −0.926715754564195996924603331143,
0.926715754564195996924603331143, 1.95567073598448482222279023681, 2.86631612844463412939210052570, 3.27381985666804460223319703873, 3.70217639365433698551581447714, 4.68022649879253263009993418914, 5.11227132195889762323685903220, 5.73530160989426067882261609281, 6.08718240220327616520154502572, 6.66019819116288679958563294672, 7.60852684015633966980596452192, 7.63173684139913449353926662917, 8.132665807481574488057539990847, 8.585866889119459033715626592235, 9.041524989863110448096712863922
