| L(s) = 1 | − 2·3-s − 4·5-s + 9-s + 2·11-s + 8·15-s + 6·23-s + 11·25-s + 4·27-s − 8·31-s − 4·33-s + 2·37-s − 4·45-s + 2·47-s + 8·49-s − 14·53-s − 8·55-s − 8·59-s + 14·67-s − 12·69-s − 22·75-s − 11·81-s + 8·89-s + 16·93-s − 18·97-s + 2·99-s + 10·103-s − 4·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s + 2.06·15-s + 1.25·23-s + 11/5·25-s + 0.769·27-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.596·45-s + 0.291·47-s + 8/7·49-s − 1.92·53-s − 1.07·55-s − 1.04·59-s + 1.71·67-s − 1.44·69-s − 2.54·75-s − 1.22·81-s + 0.847·89-s + 1.65·93-s − 1.82·97-s + 0.201·99-s + 0.985·103-s − 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.56872923108689049659304407235, −7.11887939596365119714029630413, −6.82417552681278509891696815650, −6.40406600892807193601190711266, −5.90916864692261659322807695472, −5.33889353061379062549921581920, −4.98760820845825421991059883086, −4.55861072129707426547924929395, −4.06610872348860958992456877447, −3.61627325378515210550297540208, −3.16500415333424153504767225991, −2.53515393119178065883940835809, −1.46622120430306728422962260467, −0.78440829048920899994600374863, 0,
0.78440829048920899994600374863, 1.46622120430306728422962260467, 2.53515393119178065883940835809, 3.16500415333424153504767225991, 3.61627325378515210550297540208, 4.06610872348860958992456877447, 4.55861072129707426547924929395, 4.98760820845825421991059883086, 5.33889353061379062549921581920, 5.90916864692261659322807695472, 6.40406600892807193601190711266, 6.82417552681278509891696815650, 7.11887939596365119714029630413, 7.56872923108689049659304407235
