| L(s) = 1 | + 2·7-s − 9-s − 12·23-s + 10·25-s − 16·31-s − 24·41-s − 24·47-s + 3·49-s − 2·63-s + 12·71-s + 20·73-s + 8·79-s + 81-s − 24·89-s − 20·97-s + 16·103-s − 12·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 24·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s − 2.50·23-s + 2·25-s − 2.87·31-s − 3.74·41-s − 3.50·47-s + 3/7·49-s − 0.251·63-s + 1.42·71-s + 2.34·73-s + 0.900·79-s + 1/9·81-s − 2.54·89-s − 2.03·97-s + 1.57·103-s − 1.12·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.89·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.983081919844388091019770191939, −7.942696774340037832322275123907, −7.16871189149650957205261704801, −6.89595211369618080201968592524, −6.60145928989985901456930619813, −6.39204791689497160412699890303, −5.57844285564960434267849504143, −5.54229606706781697400865913072, −5.02742272292131235884624338243, −4.87548881225291001669457992762, −4.37511585051108608285918087555, −3.72163334974701637072978281481, −3.46445598051237037944234230449, −3.32011788639021068020984106117, −2.44472394532890852593298668173, −2.06875207797482378266165752435, −1.64127100441604991309627763858, −1.29227534929850042669879402250, 0, 0,
1.29227534929850042669879402250, 1.64127100441604991309627763858, 2.06875207797482378266165752435, 2.44472394532890852593298668173, 3.32011788639021068020984106117, 3.46445598051237037944234230449, 3.72163334974701637072978281481, 4.37511585051108608285918087555, 4.87548881225291001669457992762, 5.02742272292131235884624338243, 5.54229606706781697400865913072, 5.57844285564960434267849504143, 6.39204791689497160412699890303, 6.60145928989985901456930619813, 6.89595211369618080201968592524, 7.16871189149650957205261704801, 7.942696774340037832322275123907, 7.983081919844388091019770191939
