| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s − 12·23-s + 8·25-s − 4·29-s − 6·32-s − 20·37-s − 16·43-s + 6·44-s + 24·46-s − 16·50-s − 16·53-s + 8·58-s + 7·64-s + 4·67-s + 4·71-s + 40·74-s + 32·79-s + 32·86-s − 8·88-s − 36·92-s + 24·100-s + 32·106-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 − 2.50·23-s + 8/5·25-s − 0.742·29-s − 1.06·32-s − 3.28·37-s − 2.43·43-s + 0.904·44-s + 3.53·46-s − 2.26·50-s − 2.19·53-s + 1.05·58-s + 7/8·64-s + 0.488·67-s + 0.474·71-s + 4.64·74-s + 3.60·79-s + 3.45·86-s − 0.852·88-s − 3.75·92-s + 12/5·100-s + 3.10·106-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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.53920741335489587690917938226, −7.12053203459857555400599003846, −6.93173826762667047360434676102, −6.65268893283410813193648402452, −6.17381493138969563471037964906, −6.15645615616629206463948287474, −5.42537264291110908041274093369, −5.23903062470301943226404285557, −4.78659205850803489284580643843, −4.44841502794501465897132187689, −3.67163113774065709370095481464, −3.48398593488093022778896486393, −3.39699600197380146673885232977, −2.67502100629998347364673387373, −2.01619476394609927902420789822, −1.91594254415635647566685483353, −1.52028266885408704269461303539, −0.925726703647732877729297116843, 0, 0,
0.925726703647732877729297116843, 1.52028266885408704269461303539, 1.91594254415635647566685483353, 2.01619476394609927902420789822, 2.67502100629998347364673387373, 3.39699600197380146673885232977, 3.48398593488093022778896486393, 3.67163113774065709370095481464, 4.44841502794501465897132187689, 4.78659205850803489284580643843, 5.23903062470301943226404285557, 5.42537264291110908041274093369, 6.15645615616629206463948287474, 6.17381493138969563471037964906, 6.65268893283410813193648402452, 6.93173826762667047360434676102, 7.12053203459857555400599003846, 7.53920741335489587690917938226
