| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 5·7-s + 4·8-s + 4·10-s + 2·11-s − 5·13-s − 10·14-s + 8·16-s + 2·17-s + 4·20-s + 4·22-s + 6·23-s + 3·25-s − 10·26-s − 10·28-s − 4·29-s − 14·31-s + 8·32-s + 4·34-s − 10·35-s + 2·37-s + 8·40-s + 6·41-s − 43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 1.88·7-s + 1.41·8-s + 1.26·10-s + 0.603·11-s − 1.38·13-s − 2.67·14-s + 2·16-s + 0.485·17-s + 0.894·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s − 1.96·26-s − 1.88·28-s − 0.742·29-s − 2.51·31-s + 1.41·32-s + 0.685·34-s − 1.69·35-s + 0.328·37-s + 1.26·40-s + 0.937·41-s − 0.152·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.120562529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120562529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.18980137620173473128583712908, −10.58466704533406378544303841166, −9.841940536376980381046873061917, −9.744780878214181370781678972733, −9.388679003272540244048190508296, −8.983216747599238767323641699146, −8.210207343397531711686023667978, −7.38678411475945676825566959440, −7.22891461842650357977994977254, −6.76994410642132475742346448226, −6.40511687785417313766058974595, −5.53210491696686668663124785236, −5.45958830313248885414808741965, −5.12588428946771664653742931642, −4.19776069111908129606904026019, −3.65629075785898758819036233409, −3.53026721314585480121250632595, −2.43863148074310108346230017260, −2.27699844307086359268012777095, −0.949416655390928422297577573260,
0.949416655390928422297577573260, 2.27699844307086359268012777095, 2.43863148074310108346230017260, 3.53026721314585480121250632595, 3.65629075785898758819036233409, 4.19776069111908129606904026019, 5.12588428946771664653742931642, 5.45958830313248885414808741965, 5.53210491696686668663124785236, 6.40511687785417313766058974595, 6.76994410642132475742346448226, 7.22891461842650357977994977254, 7.38678411475945676825566959440, 8.210207343397531711686023667978, 8.983216747599238767323641699146, 9.388679003272540244048190508296, 9.744780878214181370781678972733, 9.841940536376980381046873061917, 10.58466704533406378544303841166, 11.18980137620173473128583712908
