| L(s) = 1 | + 2·3-s + 4-s − 2·7-s + 3·9-s + 4·11-s + 2·12-s − 3·16-s + 4·17-s + 4·19-s − 4·21-s − 8·23-s + 4·27-s − 2·28-s − 4·29-s + 12·31-s + 8·33-s + 3·36-s − 4·37-s − 4·41-s + 4·44-s − 8·47-s − 6·48-s + 3·49-s + 8·51-s + 16·53-s + 8·57-s − 4·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.755·7-s + 9-s + 1.20·11-s + 0.577·12-s − 3/4·16-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 0.769·27-s − 0.377·28-s − 0.742·29-s + 2.15·31-s + 1.39·33-s + 1/2·36-s − 0.657·37-s − 0.624·41-s + 0.603·44-s − 1.16·47-s − 0.866·48-s + 3/7·49-s + 1.12·51-s + 2.19·53-s + 1.05·57-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.141491924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.141491924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.28299398728800897049558528258, −10.39010313504787535454496522042, −9.950960302441261767004383990112, −9.868084733126321368304859486514, −9.269254249531983603936081662591, −9.022742741780680793134051674963, −8.302662917073549728740059517430, −7.995355329640417161264884598639, −7.63336443622346506727556439854, −6.78153876579360587710267337064, −6.71373771480295834420418654362, −6.28769649091711699777333965213, −5.48696862629181682043759596209, −4.97165091885064165462761536384, −4.05644795091330065926893331078, −3.75937823312199115821787225357, −3.29003001980827580184157311209, −2.52400202601718389199352352664, −2.00819675316936200586395952175, −1.05542103816238995373673142966,
1.05542103816238995373673142966, 2.00819675316936200586395952175, 2.52400202601718389199352352664, 3.29003001980827580184157311209, 3.75937823312199115821787225357, 4.05644795091330065926893331078, 4.97165091885064165462761536384, 5.48696862629181682043759596209, 6.28769649091711699777333965213, 6.71373771480295834420418654362, 6.78153876579360587710267337064, 7.63336443622346506727556439854, 7.995355329640417161264884598639, 8.302662917073549728740059517430, 9.022742741780680793134051674963, 9.269254249531983603936081662591, 9.868084733126321368304859486514, 9.950960302441261767004383990112, 10.39010313504787535454496522042, 11.28299398728800897049558528258
