| L(s) = 1 | − 2·3-s − 3·4-s + 9-s + 6·12-s + 5·16-s − 9·25-s + 4·27-s + 4·31-s − 3·36-s + 6·37-s − 10·48-s + 10·49-s − 3·64-s − 4·67-s + 18·75-s − 11·81-s − 8·93-s − 26·97-s + 27·100-s + 16·103-s − 12·108-s − 12·111-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 1/3·9-s + 1.73·12-s + 5/4·16-s − 9/5·25-s + 0.769·27-s + 0.718·31-s − 1/2·36-s + 0.986·37-s − 1.44·48-s + 10/7·49-s − 3/8·64-s − 0.488·67-s + 2.07·75-s − 1.22·81-s − 0.829·93-s − 2.63·97-s + 2.69·100-s + 1.57·103-s − 1.15·108-s − 1.13·111-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.281143958638857692960467551261, −8.634054297830232023940201708085, −8.249977975353377798009088871094, −7.75592585898703081972787164653, −7.15112546908013756778030772487, −6.48577630694408373513070381672, −5.86606203787301044654210886361, −5.64630551311784916090773833238, −5.04164591351084708065846963538, −4.41303363846825067569218781441, −4.14650031509247077700092409613, −3.35443968705909100459094716436, −2.36213852699006213104899542481, −1.05049044658149268370353805033, 0,
1.05049044658149268370353805033, 2.36213852699006213104899542481, 3.35443968705909100459094716436, 4.14650031509247077700092409613, 4.41303363846825067569218781441, 5.04164591351084708065846963538, 5.64630551311784916090773833238, 5.86606203787301044654210886361, 6.48577630694408373513070381672, 7.15112546908013756778030772487, 7.75592585898703081972787164653, 8.249977975353377798009088871094, 8.634054297830232023940201708085, 9.281143958638857692960467551261
