| L(s) = 1 | − 5-s − 4·7-s + 3·11-s + 2·13-s + 7·19-s − 5·23-s − 6·31-s + 4·35-s − 3·37-s + 6·41-s + 16·43-s − 47-s + 9·49-s + 5·53-s − 3·55-s − 4·59-s + 8·61-s − 2·65-s + 12·71-s + 14·73-s − 12·77-s + 16·79-s + 32·83-s + 6·89-s − 8·91-s − 7·95-s + 32·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.904·11-s + 0.554·13-s + 1.60·19-s − 1.04·23-s − 1.07·31-s + 0.676·35-s − 0.493·37-s + 0.937·41-s + 2.43·43-s − 0.145·47-s + 9/7·49-s + 0.686·53-s − 0.404·55-s − 0.520·59-s + 1.02·61-s − 0.248·65-s + 1.42·71-s + 1.63·73-s − 1.36·77-s + 1.80·79-s + 3.51·83-s + 0.635·89-s − 0.838·91-s − 0.718·95-s + 3.24·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.348209209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.348209209\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−9.260265202689458243795270816822, −8.902864024088460803182879394994, −8.301940240250920807783633374213, −7.891368604562447538540632708795, −7.47426403077949028117787254963, −7.33266499804856655973525924971, −6.71230022867760360007390635906, −6.43816277742489665642610738531, −5.93250451670301538326087697393, −5.88581021447571321201622606447, −5.14080045295102106053661375082, −4.85570130901002677467830035324, −3.96807218411655335046076459677, −3.89184152053049030664057411143, −3.36831930887904137204450785417, −3.28250322732277928044848727034, −2.17761410624789828117848272655, −2.17116471879093239034055719207, −0.831904075491864375468347720537, −0.74055757516995056951015704646,
0.74055757516995056951015704646, 0.831904075491864375468347720537, 2.17116471879093239034055719207, 2.17761410624789828117848272655, 3.28250322732277928044848727034, 3.36831930887904137204450785417, 3.89184152053049030664057411143, 3.96807218411655335046076459677, 4.85570130901002677467830035324, 5.14080045295102106053661375082, 5.88581021447571321201622606447, 5.93250451670301538326087697393, 6.43816277742489665642610738531, 6.71230022867760360007390635906, 7.33266499804856655973525924971, 7.47426403077949028117787254963, 7.891368604562447538540632708795, 8.301940240250920807783633374213, 8.902864024088460803182879394994, 9.260265202689458243795270816822
