| L(s) = 1 | + 54·5-s − 610·7-s + 1.45e3·9-s + 1.06e3·11-s + 9.63e3·17-s − 1.37e4·19-s + 4.12e4·23-s + 1.56e4·25-s − 3.29e4·35-s + 1.42e5·43-s + 7.87e4·45-s + 7.51e4·47-s + 1.17e5·49-s + 5.73e4·55-s + 5.70e4·61-s − 8.89e5·63-s − 3.84e5·73-s − 6.47e5·77-s + 1.59e6·81-s − 2.26e6·83-s + 5.20e5·85-s − 7.40e5·95-s + 1.54e6·99-s + 4.12e6·101-s + 2.22e6·115-s − 5.87e6·119-s + 1.77e6·121-s + ⋯ |
| L(s) = 1 | + 0.431·5-s − 1.77·7-s + 2·9-s + 0.797·11-s + 1.96·17-s − 2·19-s + 3.38·23-s + 25-s − 0.768·35-s + 1.79·43-s + 0.863·45-s + 0.723·47-s + 49-s + 0.344·55-s + 0.251·61-s − 3.55·63-s − 0.987·73-s − 1.41·77-s + 3·81-s − 3.95·83-s + 0.846·85-s − 0.863·95-s + 1.59·99-s + 3.99·101-s + 1.46·115-s − 3.48·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.425152858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.425152858\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 54 T - 12709 T^{2} - 54 p^{6} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 610 T + 254451 T^{2} + 610 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1062 T - 643717 T^{2} - 1062 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9630 T + 68599331 T^{2} - 9630 p^{6} T^{3} + p^{12} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 20610 T + p^{6} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 142630 T + 14021953851 T^{2} - 142630 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 75150 T - 5131692829 T^{2} - 75150 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 57062 T - 48264302517 T^{2} - 57062 p^{6} T^{3} + p^{12} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 384050 T - 3839823789 T^{2} + 384050 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 1131030 T + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−13.13916985873738827355526648914, −12.92910546163084820562973390869, −12.70754786026471347494560994465, −12.34481914792019523880085169072, −11.23558566226203095796494135437, −10.61822261252683773305207049687, −10.06853510676634367014272637249, −9.812378182070511461324973175288, −8.989512806332245176350614798988, −8.840090718763794422822185015426, −7.31121136206491797928879461349, −7.20578489496551891896268427215, −6.51534767134144540042328342593, −6.00011154716434868380363394154, −4.95277761462219694625711014914, −4.20333397752856320841372488735, −3.42926797368717717599525481559, −2.71469019291539569245356734082, −1.34520879767376708100483109948, −0.818571483744313695778946047792,
0.818571483744313695778946047792, 1.34520879767376708100483109948, 2.71469019291539569245356734082, 3.42926797368717717599525481559, 4.20333397752856320841372488735, 4.95277761462219694625711014914, 6.00011154716434868380363394154, 6.51534767134144540042328342593, 7.20578489496551891896268427215, 7.31121136206491797928879461349, 8.840090718763794422822185015426, 8.989512806332245176350614798988, 9.812378182070511461324973175288, 10.06853510676634367014272637249, 10.61822261252683773305207049687, 11.23558566226203095796494135437, 12.34481914792019523880085169072, 12.70754786026471347494560994465, 12.92910546163084820562973390869, 13.13916985873738827355526648914
