| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s − 12·29-s + 36-s + 4·41-s − 8·44-s + 14·49-s + 12·61-s − 64-s + 24·71-s − 8·79-s + 81-s + 12·89-s − 8·99-s + 12·101-s + 4·109-s + 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 2.22·29-s + 1/6·36-s + 0.624·41-s − 1.20·44-s + 2·49-s + 1.53·61-s − 1/8·64-s + 2.84·71-s − 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.804·99-s + 1.19·101-s + 0.383·109-s + 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.766328673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766328673\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.823623606509020656478837403945, −8.652073031220198544686309691844, −7.88507659192179538204636317096, −7.84073788992424412986184550236, −7.15451949686502256222816543093, −7.02433579955803850688325630552, −6.41143786774386183294768552084, −6.32255194420537303970919936655, −5.58614307487240771166760417918, −5.57670475055006636379217977144, −5.00711718915226518335271932910, −4.43442479441212564041872598842, −3.93656014834502656007336444840, −3.85481163355579872014361968482, −3.50941999520465034312702353279, −2.82651480245653150986885647515, −2.10717368099693465253571325368, −1.82895585722543279937480641376, −1.03805984582886654009616912223, −0.60304471300815525745166106558,
0.60304471300815525745166106558, 1.03805984582886654009616912223, 1.82895585722543279937480641376, 2.10717368099693465253571325368, 2.82651480245653150986885647515, 3.50941999520465034312702353279, 3.85481163355579872014361968482, 3.93656014834502656007336444840, 4.43442479441212564041872598842, 5.00711718915226518335271932910, 5.57670475055006636379217977144, 5.58614307487240771166760417918, 6.32255194420537303970919936655, 6.41143786774386183294768552084, 7.02433579955803850688325630552, 7.15451949686502256222816543093, 7.84073788992424412986184550236, 7.88507659192179538204636317096, 8.652073031220198544686309691844, 8.823623606509020656478837403945
