| L(s) = 1 | − 256·4-s + 9.09e3·9-s + 5.79e4·11-s + 6.55e4·16-s + 5.91e5·19-s + 7.44e6·29-s + 4.67e6·31-s − 2.32e6·36-s + 4.31e7·41-s − 1.48e7·44-s + 5.90e7·49-s − 3.23e7·59-s − 8.78e7·61-s − 1.67e7·64-s + 3.22e8·71-s − 1.51e8·76-s + 1.16e9·79-s − 3.04e8·81-s − 9.40e8·89-s + 5.27e8·99-s − 1.72e8·101-s + 1.20e9·109-s − 1.90e9·116-s − 2.19e9·121-s − 1.19e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.461·9-s + 1.19·11-s + 1/4·16-s + 1.04·19-s + 1.95·29-s + 0.908·31-s − 0.230·36-s + 2.38·41-s − 0.597·44-s + 1.46·49-s − 0.347·59-s − 0.812·61-s − 1/8·64-s + 1.50·71-s − 0.520·76-s + 3.37·79-s − 0.786·81-s − 1.58·89-s + 0.551·99-s − 0.164·101-s + 0.820·109-s − 0.977·116-s − 0.930·121-s − 0.454·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.514890307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.514890307\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 1010 p^{2} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 59010250 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 28992 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 5833488170 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 116637458690 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 295780 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2872347954230 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3722970 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2335772 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 142408817175430 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 21593862 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 887846630571250 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2211509934714490 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3039721203142010 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 16162860 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 43928158 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 47761455709303810 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 161307732 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56661056318598670 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 583345720 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 373668173587851010 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 470133690 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1506576214182604990 T^{2} + p^{18} T^{4} \) |
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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
−13.76121509391862612632516365355, −13.63208259089429758720500353148, −12.43850776240441290540378319827, −12.36236405542627047038927028266, −11.67632425103022843162697652038, −10.92424109547119650085399219459, −10.31275631262194042080541805424, −9.503795441435983635435565942511, −9.317160356456233877177730574768, −8.445057283056486400140726836587, −7.83089663528276062315346178643, −7.03244260041193770465922104639, −6.39716814557642448131544843936, −5.65532206274870606920693094746, −4.67871740542370016467871817678, −4.19783692777209580522825917770, −3.32331821745383815194276435973, −2.38886327994890721359874780356, −1.11282376535141178738538947066, −0.796891934557313161269382118841,
0.796891934557313161269382118841, 1.11282376535141178738538947066, 2.38886327994890721359874780356, 3.32331821745383815194276435973, 4.19783692777209580522825917770, 4.67871740542370016467871817678, 5.65532206274870606920693094746, 6.39716814557642448131544843936, 7.03244260041193770465922104639, 7.83089663528276062315346178643, 8.445057283056486400140726836587, 9.317160356456233877177730574768, 9.503795441435983635435565942511, 10.31275631262194042080541805424, 10.92424109547119650085399219459, 11.67632425103022843162697652038, 12.36236405542627047038927028266, 12.43850776240441290540378319827, 13.63208259089429758720500353148, 13.76121509391862612632516365355
