| L(s) = 1 | + 3·2-s + 5·4-s + 5·8-s + 9-s + 3·11-s + 16-s + 3·18-s + 9·22-s + 6·23-s − 4·25-s − 14·29-s − 7·32-s + 5·36-s − 37-s − 43-s + 15·44-s + 18·46-s − 12·50-s + 5·53-s − 42·58-s − 15·64-s + 22·67-s − 13·71-s + 5·72-s − 3·74-s − 2·79-s − 8·81-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 5/2·4-s + 1.76·8-s + 1/3·9-s + 0.904·11-s + 1/4·16-s + 0.707·18-s + 1.91·22-s + 1.25·23-s − 4/5·25-s − 2.59·29-s − 1.23·32-s + 5/6·36-s − 0.164·37-s − 0.152·43-s + 2.26·44-s + 2.65·46-s − 1.69·50-s + 0.686·53-s − 5.51·58-s − 1.87·64-s + 2.68·67-s − 1.54·71-s + 0.589·72-s − 0.348·74-s − 0.225·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52822 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52822 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.316152011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.316152011\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 - p T + p T^{2} ) \) |
| 7 | | \( 1 \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−10.19088637243585327023143286754, −9.436557221532260673465592546759, −9.224177946359878073969986007743, −8.514941746548024135577971964819, −7.70913023103413429527256496778, −7.07574695899204698994496466977, −6.83312379534503841566957499846, −6.08084201815418003788055359385, −5.60193839500756341970255558117, −5.16223876078519798724590118138, −4.44958536623122435193721934457, −3.82576935617193731959842165725, −3.56059784290201737401318904489, −2.60626038540882664448149621670, −1.70155724670327511287010271607,
1.70155724670327511287010271607, 2.60626038540882664448149621670, 3.56059784290201737401318904489, 3.82576935617193731959842165725, 4.44958536623122435193721934457, 5.16223876078519798724590118138, 5.60193839500756341970255558117, 6.08084201815418003788055359385, 6.83312379534503841566957499846, 7.07574695899204698994496466977, 7.70913023103413429527256496778, 8.514941746548024135577971964819, 9.224177946359878073969986007743, 9.436557221532260673465592546759, 10.19088637243585327023143286754
