| L(s) = 1 | − 2·2-s − 3·3-s + 14·5-s + 6·6-s + 7·7-s + 8·8-s − 28·10-s + 16·11-s + 91·13-s − 14·14-s − 42·15-s − 16·16-s + 33·17-s − 144·19-s − 21·21-s − 32·22-s + 44·23-s − 24·24-s − 103·25-s − 182·26-s + 27·27-s + 185·29-s + 84·30-s + 368·31-s − 48·33-s − 66·34-s + 98·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1.25·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 0.885·10-s + 0.438·11-s + 1.94·13-s − 0.267·14-s − 0.722·15-s − 1/4·16-s + 0.470·17-s − 1.73·19-s − 0.218·21-s − 0.310·22-s + 0.398·23-s − 0.204·24-s − 0.823·25-s − 1.37·26-s + 0.192·27-s + 1.18·29-s + 0.511·30-s + 2.13·31-s − 0.253·33-s − 0.332·34-s + 0.473·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.746665803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.746665803\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 16 T - 1075 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T - 3824 T^{2} - 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 144 T + 13877 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T - 10231 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 185 T + 9836 T^{2} - 185 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 184 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 225 T - 28 T^{2} - 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 165 T - 41696 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 20 T - 79107 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 111 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 220 T - 156979 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 269 T - 154620 T^{2} - 269 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 700 T + 189237 T^{2} + 700 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1016 T + 674345 T^{2} + 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 947 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1140 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 858 T + 31195 T^{2} + 858 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 862 T - 169629 T^{2} - 862 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.43837775708358076768877769786, −10.32831874177632775759983687995, −9.664783565979109293072630849973, −9.396291977999296529396676013556, −8.824507694262251561154497301593, −8.505357431905524333757213264903, −7.949197664810170741502295954236, −7.82984777783604758128515695779, −6.63819955595262935515369786913, −6.48244244290800971753884589590, −6.10284160802528264449815421169, −5.82121769276327689602871501259, −5.00508627888155321034310114847, −4.55608632008029551678972361576, −3.98418452603994475671837232896, −3.32184788922733943231276359733, −2.34263532179906412086196596470, −1.91311040720045177390579711415, −1.04142365834823197438872087429, −0.73986929017033639339357884171,
0.73986929017033639339357884171, 1.04142365834823197438872087429, 1.91311040720045177390579711415, 2.34263532179906412086196596470, 3.32184788922733943231276359733, 3.98418452603994475671837232896, 4.55608632008029551678972361576, 5.00508627888155321034310114847, 5.82121769276327689602871501259, 6.10284160802528264449815421169, 6.48244244290800971753884589590, 6.63819955595262935515369786913, 7.82984777783604758128515695779, 7.949197664810170741502295954236, 8.505357431905524333757213264903, 8.824507694262251561154497301593, 9.396291977999296529396676013556, 9.664783565979109293072630849973, 10.32831874177632775759983687995, 10.43837775708358076768877769786
