| L(s) = 1 | − 6·7-s − 3·9-s − 6·17-s − 16·23-s − 25-s + 2·31-s + 8·41-s + 24·47-s + 13·49-s + 18·63-s − 6·71-s + 20·73-s − 16·79-s + 30·89-s − 12·97-s − 8·103-s − 16·113-s + 36·119-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 9-s − 1.45·17-s − 3.33·23-s − 1/5·25-s + 0.359·31-s + 1.24·41-s + 3.50·47-s + 13/7·49-s + 2.26·63-s − 0.712·71-s + 2.34·73-s − 1.80·79-s + 3.17·89-s − 1.21·97-s − 0.788·103-s − 1.50·113-s + 3.30·119-s − 0.0909·121-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 + 1.45·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.740546603057181807898507357560, −7.909551363410651889764112586006, −7.77270660990524444649898022305, −7.20748501929378748583051255191, −6.82626916871552478550077794932, −6.37419056431625582475047682117, −6.09535469577282785154569942488, −6.00197685552789543046916491073, −5.58330832547238990472513382801, −5.04465345149384411683308847517, −4.27437182921435493887964887633, −3.95539993664598257024996455228, −3.91382132591281820584977839093, −3.18142779054197125994179849725, −2.76254692520634430098911034900, −2.23728560771767265218240020092, −2.16330794505844415174798995602, −0.925036737731665753943887119321, 0, 0,
0.925036737731665753943887119321, 2.16330794505844415174798995602, 2.23728560771767265218240020092, 2.76254692520634430098911034900, 3.18142779054197125994179849725, 3.91382132591281820584977839093, 3.95539993664598257024996455228, 4.27437182921435493887964887633, 5.04465345149384411683308847517, 5.58330832547238990472513382801, 6.00197685552789543046916491073, 6.09535469577282785154569942488, 6.37419056431625582475047682117, 6.82626916871552478550077794932, 7.20748501929378748583051255191, 7.77270660990524444649898022305, 7.909551363410651889764112586006, 8.740546603057181807898507357560
