| L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s + 2·9-s + 3·14-s + 16-s + 2·17-s − 2·18-s − 5·23-s − 2·25-s − 3·28-s + 31-s − 32-s − 2·34-s + 2·36-s + 17·41-s + 5·46-s + 8·47-s + 5·49-s + 2·50-s + 3·56-s − 62-s − 6·63-s + 64-s + 2·68-s − 11·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s + 2/3·9-s + 0.801·14-s + 1/4·16-s + 0.485·17-s − 0.471·18-s − 1.04·23-s − 2/5·25-s − 0.566·28-s + 0.179·31-s − 0.176·32-s − 0.342·34-s + 1/3·36-s + 2.65·41-s + 0.737·46-s + 1.16·47-s + 5/7·49-s + 0.282·50-s + 0.400·56-s − 0.127·62-s − 0.755·63-s + 1/8·64-s + 0.242·68-s − 1.30·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9554430931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9554430931\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 7 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.452776506099791675488433550861, −8.966399061351815426167779607630, −8.442066348373644945412995600007, −7.76076108729885933576340127571, −7.48510701666837172145886467956, −7.01178990457040861318470475888, −6.38494480225536504645148083584, −5.88347248279884337463631045385, −5.62910930364000478881515357287, −4.50024261354158604965968802670, −4.09692248452966152942490981680, −3.37061023370712022631398948022, −2.69422267066174424314711114697, −1.91793348953003849996031552383, −0.75444864370146818047565933462,
0.75444864370146818047565933462, 1.91793348953003849996031552383, 2.69422267066174424314711114697, 3.37061023370712022631398948022, 4.09692248452966152942490981680, 4.50024261354158604965968802670, 5.62910930364000478881515357287, 5.88347248279884337463631045385, 6.38494480225536504645148083584, 7.01178990457040861318470475888, 7.48510701666837172145886467956, 7.76076108729885933576340127571, 8.442066348373644945412995600007, 8.966399061351815426167779607630, 9.452776506099791675488433550861
