| L(s) = 1 | + 4.03e3·3-s − 1.56e5·5-s + 1.64e6·7-s + 4.78e6·9-s + 3.73e7·11-s + 6.52e7·13-s − 6.29e8·15-s + 6.26e8·17-s + 6.63e9·21-s + 1.83e10·25-s − 7.65e9·27-s + 9.77e9·29-s + 1.50e11·33-s − 2.57e11·35-s + 2.63e11·39-s − 7.47e11·45-s − 7.19e11·47-s + 2.03e12·49-s + 2.52e12·51-s − 5.84e12·55-s + 7.87e12·63-s − 1.01e13·65-s − 3.58e12·71-s + 4.40e13·73-s + 7.38e13·75-s + 6.15e13·77-s + 2.70e13·79-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 2·5-s + 2·7-s + 9-s + 1.91·11-s + 1.04·13-s − 3.68·15-s + 1.52·17-s + 3.68·21-s + 3·25-s − 0.732·27-s + 0.566·29-s + 3.53·33-s − 4·35-s + 1.91·39-s − 2·45-s − 1.41·47-s + 3·49-s + 2.81·51-s − 3.83·55-s + 2·63-s − 2.08·65-s − 0.393·71-s + 3.98·73-s + 5.52·75-s + 3.83·77-s + 1.41·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(12.56281624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.56281624\) |
| \(L(8)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4031 T + 11465992 T^{2} - 4031 p^{14} T^{3} + p^{28} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 37379173 T + 1017452740580688 T^{2} - 37379173 p^{14} T^{3} + p^{28} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 65279611 T + 324051226612032 T^{2} - 65279611 p^{14} T^{3} + p^{28} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 626193259 T + 223740171057640152 T^{2} - 626193259 p^{14} T^{3} + p^{28} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9775649497 T - \)\(20\!\cdots\!72\)\( T^{2} - 9775649497 p^{14} T^{3} + p^{28} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 719081600801 T + \)\(26\!\cdots\!32\)\( T^{2} + 719081600801 p^{14} T^{3} + p^{28} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 1790558995678 T + p^{14} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 22033597628414 T + p^{14} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 27088287440917 T + \)\(36\!\cdots\!08\)\( T^{2} - 27088287440917 p^{14} T^{3} + p^{28} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 33726754263974 T + p^{14} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 24587561871581 T - \)\(59\!\cdots\!08\)\( T^{2} + 24587561871581 p^{14} T^{3} + p^{28} T^{4} \) |
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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
−10.96685368291919863133670654478, −10.46190700518569773880775033820, −9.281571959389704049984441290064, −9.235683580456157871659085428469, −8.393595335618482831905710801502, −8.372122657643408946378519209626, −7.81378931507096473582493330217, −7.79207484804596396492581624957, −6.87690356099505958393059512545, −6.39027526118145245186853055150, −5.30698294639320084163440266090, −4.83680651959448310358059411737, −4.07384295373710895020554154382, −3.83163141540014169527566699269, −3.41393011199270529773883905694, −2.97963605078784317409618736364, −2.06145674629177102002093987739, −1.58701245348125164774649471807, −0.827022453829793727969804378848, −0.814437807283211062953761229512,
0.814437807283211062953761229512, 0.827022453829793727969804378848, 1.58701245348125164774649471807, 2.06145674629177102002093987739, 2.97963605078784317409618736364, 3.41393011199270529773883905694, 3.83163141540014169527566699269, 4.07384295373710895020554154382, 4.83680651959448310358059411737, 5.30698294639320084163440266090, 6.39027526118145245186853055150, 6.87690356099505958393059512545, 7.79207484804596396492581624957, 7.81378931507096473582493330217, 8.372122657643408946378519209626, 8.393595335618482831905710801502, 9.235683580456157871659085428469, 9.281571959389704049984441290064, 10.46190700518569773880775033820, 10.96685368291919863133670654478
