| L(s) = 1 | − 3·3-s − 4-s − 9·7-s + 6·9-s + 3·12-s − 13·13-s + 16-s − 12·19-s + 27·21-s − 8·25-s − 9·27-s + 9·28-s − 8·31-s − 6·36-s − 5·37-s + 39·39-s − 43-s − 3·48-s + 47·49-s + 13·52-s + 36·57-s − 3·61-s − 54·63-s − 64-s − 16·67-s + 8·73-s + 24·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s − 3.40·7-s + 2·9-s + 0.866·12-s − 3.60·13-s + 1/4·16-s − 2.75·19-s + 5.89·21-s − 8/5·25-s − 1.73·27-s + 1.70·28-s − 1.43·31-s − 36-s − 0.821·37-s + 6.24·39-s − 0.152·43-s − 0.433·48-s + 47/7·49-s + 1.80·52-s + 4.76·57-s − 0.384·61-s − 6.80·63-s − 1/8·64-s − 1.95·67-s + 0.936·73-s + 2.77·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 824508 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 824508 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 619 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.28434166450402902423422046946, −6.98463328609589548695164243736, −6.65086187770472139923030681625, −6.15698642089545970849232726020, −5.87329940180443475282818538472, −5.43029414079246726301275165236, −4.78652667752701927044417566764, −4.37260448533769623930706227834, −3.89602610304574057223439346219, −3.28074992137089612912847999132, −2.52048941255719544362049896044, −2.08728394529744675492547926907, 0, 0, 0,
2.08728394529744675492547926907, 2.52048941255719544362049896044, 3.28074992137089612912847999132, 3.89602610304574057223439346219, 4.37260448533769623930706227834, 4.78652667752701927044417566764, 5.43029414079246726301275165236, 5.87329940180443475282818538472, 6.15698642089545970849232726020, 6.65086187770472139923030681625, 6.98463328609589548695164243736, 7.28434166450402902423422046946
