| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 8·11-s + 5·16-s + 16·22-s + 16·23-s − 8·25-s + 4·29-s + 6·32-s + 8·37-s − 8·43-s + 24·44-s + 32·46-s − 16·50-s + 8·53-s + 8·58-s + 7·64-s − 24·67-s + 16·74-s − 32·79-s − 16·86-s + 32·88-s + 48·92-s − 24·100-s + 16·106-s − 8·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2.41·11-s + 5/4·16-s + 3.41·22-s + 3.33·23-s − 8/5·25-s + 0.742·29-s + 1.06·32-s + 1.31·37-s − 1.21·43-s + 3.61·44-s + 4.71·46-s − 2.26·50-s + 1.09·53-s + 1.05·58-s + 7/8·64-s − 2.93·67-s + 1.85·74-s − 3.60·79-s − 1.72·86-s + 3.41·88-s + 5.00·92-s − 2.39·100-s + 1.55·106-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.792408240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.792408240\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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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
−10.35138578049123002607309059887, −10.04445720000731360600711286270, −9.502518229113952405387627729243, −8.986351798200098119872455168136, −8.835374543565323964485863950256, −8.276788172758277357228894987285, −7.37849654061162292619866898500, −7.30771536769475545877594253009, −6.82490124138223785858770870706, −6.33446421236280350188296302066, −6.06068699843142175191593109764, −5.55459911588512162527671164444, −4.84241153286198224609438197456, −4.61622541306224796762577273579, −3.97821097725850109395275477104, −3.70640505808144981020585504291, −2.97412650936147419059268641675, −2.64428853613751625658277650408, −1.46445956865435966907113938437, −1.24826361104900586024417080600,
1.24826361104900586024417080600, 1.46445956865435966907113938437, 2.64428853613751625658277650408, 2.97412650936147419059268641675, 3.70640505808144981020585504291, 3.97821097725850109395275477104, 4.61622541306224796762577273579, 4.84241153286198224609438197456, 5.55459911588512162527671164444, 6.06068699843142175191593109764, 6.33446421236280350188296302066, 6.82490124138223785858770870706, 7.30771536769475545877594253009, 7.37849654061162292619866898500, 8.276788172758277357228894987285, 8.835374543565323964485863950256, 8.986351798200098119872455168136, 9.502518229113952405387627729243, 10.04445720000731360600711286270, 10.35138578049123002607309059887
