| L(s) = 1 | − 9-s + 8·19-s − 12·29-s − 16·31-s − 20·41-s − 49-s − 8·59-s − 20·61-s + 8·71-s − 16·79-s + 81-s − 12·89-s + 20·101-s − 28·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 8·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.83·19-s − 2.22·29-s − 2.87·31-s − 3.12·41-s − 1/7·49-s − 1.04·59-s − 2.56·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 1.99·101-s − 2.68·109-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s + 1.69·169-s − 0.611·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.064861520757445797939535823241, −7.79276907826150006981993084253, −7.47939955074574041020294383822, −7.30429705951522914043457123532, −6.67227472830029634070209693639, −6.59445701614279418311597421393, −5.73241181607972691553633770643, −5.68879767997465844204741802143, −5.25933633419754947274728934243, −5.06685047426033120465002119791, −4.46479175726322757361838560151, −3.84211254162130177098868463801, −3.59148924990920761356962535705, −3.22932194731174226854815383465, −2.85829013442355309935270011728, −2.09673938451124290436619254542, −1.49648659738593880773446478805, −1.47736893272648551878602418355, 0, 0,
1.47736893272648551878602418355, 1.49648659738593880773446478805, 2.09673938451124290436619254542, 2.85829013442355309935270011728, 3.22932194731174226854815383465, 3.59148924990920761356962535705, 3.84211254162130177098868463801, 4.46479175726322757361838560151, 5.06685047426033120465002119791, 5.25933633419754947274728934243, 5.68879767997465844204741802143, 5.73241181607972691553633770643, 6.59445701614279418311597421393, 6.67227472830029634070209693639, 7.30429705951522914043457123532, 7.47939955074574041020294383822, 7.79276907826150006981993084253, 8.064861520757445797939535823241
