| L(s) = 1 | + 3·4-s − 2·5-s + 12·11-s + 5·16-s + 12·19-s − 6·20-s − 25-s − 4·29-s − 20·31-s − 4·41-s + 36·44-s − 49-s − 24·55-s − 16·59-s − 4·61-s + 3·64-s − 20·71-s + 36·76-s − 8·79-s − 10·80-s + 12·89-s − 24·95-s − 3·100-s + 12·101-s − 4·109-s − 12·116-s + 86·121-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 3.61·11-s + 5/4·16-s + 2.75·19-s − 1.34·20-s − 1/5·25-s − 0.742·29-s − 3.59·31-s − 0.624·41-s + 5.42·44-s − 1/7·49-s − 3.23·55-s − 2.08·59-s − 0.512·61-s + 3/8·64-s − 2.37·71-s + 4.12·76-s − 0.900·79-s − 1.11·80-s + 1.27·89-s − 2.46·95-s − 0.299·100-s + 1.19·101-s − 0.383·109-s − 1.11·116-s + 7.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.364415880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.364415880\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−11.68149172021698514537307669320, −11.65900402784438336251184378974, −11.15682754621204080457739790913, −10.83782949321453739281959390784, −9.958907356148204600212707497163, −9.448295105735178540927949522834, −8.988335577470060659572922989961, −8.960783065017993763415907324230, −7.61379270299440107768625972683, −7.59570723637057592816899295658, −7.07117570942000618842025239364, −6.77502665514268466800942486071, −5.91216171115806701944249425884, −5.84277041949752518384985221923, −4.78965586740336360807090747123, −3.88583663336720952404155266403, −3.57134491131064569823169274652, −3.18346466442169681575130094017, −1.63786604545577669190033940755, −1.48127063129262097043069794043,
1.48127063129262097043069794043, 1.63786604545577669190033940755, 3.18346466442169681575130094017, 3.57134491131064569823169274652, 3.88583663336720952404155266403, 4.78965586740336360807090747123, 5.84277041949752518384985221923, 5.91216171115806701944249425884, 6.77502665514268466800942486071, 7.07117570942000618842025239364, 7.59570723637057592816899295658, 7.61379270299440107768625972683, 8.960783065017993763415907324230, 8.988335577470060659572922989961, 9.448295105735178540927949522834, 9.958907356148204600212707497163, 10.83782949321453739281959390784, 11.15682754621204080457739790913, 11.65900402784438336251184378974, 11.68149172021698514537307669320
