| L(s) = 1 | + 3-s + 9-s − 3·11-s − 10·17-s + 7·19-s + 6·25-s + 27-s − 3·33-s − 2·41-s − 12·43-s + 49-s − 10·51-s + 7·57-s + 7·59-s + 7·67-s + 7·73-s + 6·75-s + 81-s − 2·83-s − 12·89-s + 16·97-s − 3·99-s − 17·107-s + 25·113-s − 15·121-s − 2·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s − 2.42·17-s + 1.60·19-s + 6/5·25-s + 0.192·27-s − 0.522·33-s − 0.312·41-s − 1.82·43-s + 1/7·49-s − 1.40·51-s + 0.927·57-s + 0.911·59-s + 0.855·67-s + 0.819·73-s + 0.692·75-s + 1/9·81-s − 0.219·83-s − 1.27·89-s + 1.62·97-s − 0.301·99-s − 1.64·107-s + 2.35·113-s − 1.36·121-s − 0.180·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1783296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.017808469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.017808469\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.932199708251548550528432778136, −7.26225667673286219457230373700, −7.05943921682702913649841842517, −6.62980510433158209534053843838, −6.27319598101444636888427686355, −5.50458995713945286093625852279, −5.06889674298782586672641367662, −4.88162708745059310380482047268, −4.24237617358029675309022710393, −3.74413130046278426621928148272, −3.11325071406841293503379090858, −2.74052378364049576829420185146, −2.18369413214924489871356844142, −1.57495982096906029461247596697, −0.55957383883855022071909612658,
0.55957383883855022071909612658, 1.57495982096906029461247596697, 2.18369413214924489871356844142, 2.74052378364049576829420185146, 3.11325071406841293503379090858, 3.74413130046278426621928148272, 4.24237617358029675309022710393, 4.88162708745059310380482047268, 5.06889674298782586672641367662, 5.50458995713945286093625852279, 6.27319598101444636888427686355, 6.62980510433158209534053843838, 7.05943921682702913649841842517, 7.26225667673286219457230373700, 7.932199708251548550528432778136
