| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 7-s − 4·8-s + 4·10-s + 5·11-s − 2·13-s − 2·14-s + 5·16-s + 9·17-s − 7·19-s − 6·20-s − 10·22-s + 3·25-s + 4·26-s + 3·28-s − 2·29-s − 2·31-s − 6·32-s − 18·34-s − 2·35-s + 4·37-s + 14·38-s + 8·40-s + 13·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s − 1.41·8-s + 1.26·10-s + 1.50·11-s − 0.554·13-s − 0.534·14-s + 5/4·16-s + 2.18·17-s − 1.60·19-s − 1.34·20-s − 2.13·22-s + 3/5·25-s + 0.784·26-s + 0.566·28-s − 0.371·29-s − 0.359·31-s − 1.06·32-s − 3.08·34-s − 0.338·35-s + 0.657·37-s + 2.27·38-s + 1.26·40-s + 2.03·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64160100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64160100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122026310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122026310\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 3 p T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 85 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 139 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 139 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 29 T + 375 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−7.895165723245932661729644664781, −7.86045837496577681006573875274, −7.31104007198127897948580858691, −7.30846091544533066465150730487, −6.63322396232587533104312197693, −6.48490188857405383225262400023, −6.03993929104132250422248720483, −5.81557727903508516896436713472, −5.16635017941134523366348135907, −4.96523645173634034338884798309, −4.24311476249203450135709271002, −4.11784095610593112703531934465, −3.52959445229971046879904846797, −3.46574351967327064259699876623, −2.64145528443223281672766344411, −2.45376598708317136913990265382, −1.81039853190613346183410129098, −1.30798552850389739123226365857, −0.982930974900222728984769009356, −0.38466946381277096091454469645,
0.38466946381277096091454469645, 0.982930974900222728984769009356, 1.30798552850389739123226365857, 1.81039853190613346183410129098, 2.45376598708317136913990265382, 2.64145528443223281672766344411, 3.46574351967327064259699876623, 3.52959445229971046879904846797, 4.11784095610593112703531934465, 4.24311476249203450135709271002, 4.96523645173634034338884798309, 5.16635017941134523366348135907, 5.81557727903508516896436713472, 6.03993929104132250422248720483, 6.48490188857405383225262400023, 6.63322396232587533104312197693, 7.30846091544533066465150730487, 7.31104007198127897948580858691, 7.86045837496577681006573875274, 7.895165723245932661729644664781
