| L(s) = 1 | + 6·5-s + 6·7-s − 3·11-s − 4·13-s + 19·25-s + 6·29-s + 36·35-s + 4·37-s + 9·41-s − 9·43-s − 12·47-s + 17·49-s − 18·55-s − 15·59-s − 8·61-s − 24·65-s + 15·67-s − 12·71-s − 22·73-s − 18·77-s − 6·79-s + 12·83-s − 24·91-s − 13·97-s − 18·101-s − 24·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 2.26·7-s − 0.904·11-s − 1.10·13-s + 19/5·25-s + 1.11·29-s + 6.08·35-s + 0.657·37-s + 1.40·41-s − 1.37·43-s − 1.75·47-s + 17/7·49-s − 2.42·55-s − 1.95·59-s − 1.02·61-s − 2.97·65-s + 1.83·67-s − 1.42·71-s − 2.57·73-s − 2.05·77-s − 0.675·79-s + 1.31·83-s − 2.51·91-s − 1.31·97-s − 1.79·101-s − 2.36·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.352979699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.352979699\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−11.07176264594324387013343194173, −11.01963095866036898035130123639, −10.29988898878329244917503481632, −10.12470483992920131287663712728, −9.578784711289756383359066857849, −9.359795178654280035191976249498, −8.458084611072879884755543794328, −8.431880234787505341214263221764, −7.62672481306386818099620522974, −7.40690618741650001866817120147, −6.48228486428504359883928337938, −6.17499877304241357026965110457, −5.49725288943448370758910298446, −5.21924573534387190387217921712, −4.75018633803456829721590010696, −4.44532784685833394022858043204, −2.82790684700660830506267903358, −2.59467094678548112024010140714, −1.65220790002750827038030857601, −1.57542735696016145572239816235,
1.57542735696016145572239816235, 1.65220790002750827038030857601, 2.59467094678548112024010140714, 2.82790684700660830506267903358, 4.44532784685833394022858043204, 4.75018633803456829721590010696, 5.21924573534387190387217921712, 5.49725288943448370758910298446, 6.17499877304241357026965110457, 6.48228486428504359883928337938, 7.40690618741650001866817120147, 7.62672481306386818099620522974, 8.431880234787505341214263221764, 8.458084611072879884755543794328, 9.359795178654280035191976249498, 9.578784711289756383359066857849, 10.12470483992920131287663712728, 10.29988898878329244917503481632, 11.01963095866036898035130123639, 11.07176264594324387013343194173
