| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s + 8·23-s − 8·25-s − 6·32-s − 12·37-s − 8·43-s + 6·44-s − 16·46-s + 16·50-s − 12·53-s + 7·64-s − 20·67-s + 24·74-s − 12·79-s + 16·86-s − 8·88-s + 24·92-s − 24·100-s + 24·106-s + 8·107-s − 16·109-s + 36·113-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s + 5/4·16-s − 0.852·22-s + 1.66·23-s − 8/5·25-s − 1.06·32-s − 1.97·37-s − 1.21·43-s + 0.904·44-s − 2.35·46-s + 2.26·50-s − 1.64·53-s + 7/8·64-s − 2.44·67-s + 2.78·74-s − 1.35·79-s + 1.72·86-s − 0.852·88-s + 2.50·92-s − 2.39·100-s + 2.33·106-s + 0.773·107-s − 1.53·109-s + 3.38·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 162 T^{2} + p^{2} T^{4} \) |
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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
−7.55187761301673471673453015967, −7.26721254272450337064002970402, −6.89716801426882001128186732030, −6.70420178923516279929851793442, −6.14287740165238173076002250287, −6.06073444193383960591800673440, −5.56407894486452325254738461243, −5.18085699693998099544086586427, −4.69567332972097635517946725491, −4.52289302762779595756655046820, −3.70150398585820719141554591090, −3.63153274572928700642277542764, −2.98631160963046764950037450601, −2.93805677255495409999842716933, −2.10180590607018621017837666247, −1.84075009332519664123753407761, −1.38208657977308203308057694588, −1.04182615470983694881215387900, 0, 0,
1.04182615470983694881215387900, 1.38208657977308203308057694588, 1.84075009332519664123753407761, 2.10180590607018621017837666247, 2.93805677255495409999842716933, 2.98631160963046764950037450601, 3.63153274572928700642277542764, 3.70150398585820719141554591090, 4.52289302762779595756655046820, 4.69567332972097635517946725491, 5.18085699693998099544086586427, 5.56407894486452325254738461243, 6.06073444193383960591800673440, 6.14287740165238173076002250287, 6.70420178923516279929851793442, 6.89716801426882001128186732030, 7.26721254272450337064002970402, 7.55187761301673471673453015967
