| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 9-s − 14-s + 16-s + 3·17-s − 18-s − 6·23-s + 2·25-s + 28-s − 2·31-s − 32-s − 3·34-s + 36-s + 2·41-s + 6·46-s − 9·47-s − 11·49-s − 2·50-s − 56-s + 2·62-s + 63-s + 64-s + 3·68-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.25·23-s + 2/5·25-s + 0.188·28-s − 0.359·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.312·41-s + 0.884·46-s − 1.31·47-s − 1.57·49-s − 0.282·50-s − 0.133·56-s + 0.254·62-s + 0.125·63-s + 1/8·64-s + 0.363·68-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6620368135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6620368135\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10319314893232523985239996092, −11.51521121733675090740652413318, −11.03348393032722687486736178672, −10.34758879590260881572338401455, −9.807218831227240532497867389097, −9.399458292411099568916021535074, −8.450652361215394155377027279551, −8.089983765715588263044432499672, −7.43358708027564214212774968945, −6.71291519039905853900476391258, −5.97372862853405248766399036620, −5.15897679144909405228083945514, −4.19251140353724674055524000177, −3.11933565010302679370990991004, −1.73415606891266230708074957669,
1.73415606891266230708074957669, 3.11933565010302679370990991004, 4.19251140353724674055524000177, 5.15897679144909405228083945514, 5.97372862853405248766399036620, 6.71291519039905853900476391258, 7.43358708027564214212774968945, 8.089983765715588263044432499672, 8.450652361215394155377027279551, 9.399458292411099568916021535074, 9.807218831227240532497867389097, 10.34758879590260881572338401455, 11.03348393032722687486736178672, 11.51521121733675090740652413318, 12.10319314893232523985239996092
