| L(s) = 1 | + 3-s + 2·5-s − 4·11-s − 4·13-s + 2·15-s − 2·17-s + 4·19-s + 8·23-s + 5·25-s − 27-s + 12·29-s − 8·31-s − 4·33-s − 6·37-s − 4·39-s − 12·41-s + 8·43-s − 2·51-s + 2·53-s − 8·55-s + 4·57-s − 4·59-s + 2·61-s − 8·65-s + 4·67-s + 8·69-s + 16·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.20·11-s − 1.10·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 25-s − 0.192·27-s + 2.22·29-s − 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.640·39-s − 1.87·41-s + 1.21·43-s − 0.280·51-s + 0.274·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.992·65-s + 0.488·67-s + 0.963·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.02426693125771851321324607233, −9.483509639300566864530743596231, −9.166535085941774680575927494016, −8.859029203162157645556683985493, −8.253465339169081962557179281522, −8.180636641101783764424918200312, −7.23974721291906822564795327381, −7.19606435561194585456817772159, −6.90671324548128995301522749854, −6.14364745377578558559126635252, −5.75164118249395832507971187875, −5.08427308143170178175268646645, −4.90602183779018520341297399732, −4.71275716493911233208348817303, −3.53850914002634254838330623309, −3.27481007839388137981136165915, −2.58325468363444107765005096270, −2.39971869454034995860714342133, −1.62060026121176131173092243339, −0.67598066245933542679691704562,
0.67598066245933542679691704562, 1.62060026121176131173092243339, 2.39971869454034995860714342133, 2.58325468363444107765005096270, 3.27481007839388137981136165915, 3.53850914002634254838330623309, 4.71275716493911233208348817303, 4.90602183779018520341297399732, 5.08427308143170178175268646645, 5.75164118249395832507971187875, 6.14364745377578558559126635252, 6.90671324548128995301522749854, 7.19606435561194585456817772159, 7.23974721291906822564795327381, 8.180636641101783764424918200312, 8.253465339169081962557179281522, 8.859029203162157645556683985493, 9.166535085941774680575927494016, 9.483509639300566864530743596231, 10.02426693125771851321324607233
