| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·9-s − 2·11-s + 5·16-s + 8·18-s + 4·22-s − 4·23-s − 8·25-s − 12·29-s − 6·32-s − 12·36-s − 4·37-s + 8·43-s − 6·44-s + 8·46-s + 16·50-s − 24·53-s + 24·58-s + 7·64-s − 4·67-s − 20·71-s + 16·72-s + 8·74-s − 16·79-s + 7·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 4/3·9-s − 0.603·11-s + 5/4·16-s + 1.88·18-s + 0.852·22-s − 0.834·23-s − 8/5·25-s − 2.22·29-s − 1.06·32-s − 2·36-s − 0.657·37-s + 1.21·43-s − 0.904·44-s + 1.17·46-s + 2.26·50-s − 3.29·53-s + 3.15·58-s + 7/8·64-s − 0.488·67-s − 2.37·71-s + 1.88·72-s + 0.929·74-s − 1.80·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
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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
−9.450547251815216094657338083391, −9.411203761939149041622008443752, −8.828997911383978720063618881322, −8.540367026777806779843138756859, −7.85293374997188759826920669169, −7.84082279963447650020500724430, −7.47999686336930545781229172891, −6.88974922526010818255590206799, −6.21031904684137313513723697830, −5.97207898753295444905408185419, −5.57508637676095728193788758485, −5.18969643608982086215754502890, −4.24845863542732783150254953763, −3.83061144755951801039264188141, −2.99381486421233514984700654506, −2.80908133921127107683920355466, −1.89920185192110653583692761736, −1.60768512466675242484261785039, 0, 0,
1.60768512466675242484261785039, 1.89920185192110653583692761736, 2.80908133921127107683920355466, 2.99381486421233514984700654506, 3.83061144755951801039264188141, 4.24845863542732783150254953763, 5.18969643608982086215754502890, 5.57508637676095728193788758485, 5.97207898753295444905408185419, 6.21031904684137313513723697830, 6.88974922526010818255590206799, 7.47999686336930545781229172891, 7.84082279963447650020500724430, 7.85293374997188759826920669169, 8.540367026777806779843138756859, 8.828997911383978720063618881322, 9.411203761939149041622008443752, 9.450547251815216094657338083391
