| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 2·9-s + 8·11-s − 6·13-s − 4·15-s − 6·17-s + 4·21-s + 6·23-s − 25-s + 6·27-s + 4·29-s + 16·33-s − 4·35-s − 6·37-s − 12·39-s + 12·41-s − 6·43-s − 4·45-s + 18·47-s + 2·49-s − 12·51-s + 10·53-s − 16·55-s + 4·63-s + 12·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 2.41·11-s − 1.66·13-s − 1.03·15-s − 1.45·17-s + 0.872·21-s + 1.25·23-s − 1/5·25-s + 1.15·27-s + 0.742·29-s + 2.78·33-s − 0.676·35-s − 0.986·37-s − 1.92·39-s + 1.87·41-s − 0.914·43-s − 0.596·45-s + 2.62·47-s + 2/7·49-s − 1.68·51-s + 1.37·53-s − 2.15·55-s + 0.503·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.252134334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.252134334\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−11.81113829924476911159363322084, −11.61430812631216105871510866690, −10.90084021176686885646320573268, −10.59988447635737109397568659638, −9.872085441494781428469158973204, −9.173714116762044901093328545790, −9.080884202459432893602604602317, −8.699215331916632323856246555226, −8.194058849569125180362096735563, −7.52852954767173224025226745630, −7.00410605261467062306638326401, −6.96678563256221610647661711931, −6.14282944022681839691336982259, −5.21359479119464999923726927145, −4.38548362701850026288655848623, −4.36018674895597463202904052663, −3.67802525611135460159399783198, −2.78602271721816750954710911455, −2.23191198852958141835641901402, −1.14701040917596716806492373676,
1.14701040917596716806492373676, 2.23191198852958141835641901402, 2.78602271721816750954710911455, 3.67802525611135460159399783198, 4.36018674895597463202904052663, 4.38548362701850026288655848623, 5.21359479119464999923726927145, 6.14282944022681839691336982259, 6.96678563256221610647661711931, 7.00410605261467062306638326401, 7.52852954767173224025226745630, 8.194058849569125180362096735563, 8.699215331916632323856246555226, 9.080884202459432893602604602317, 9.173714116762044901093328545790, 9.872085441494781428469158973204, 10.59988447635737109397568659638, 10.90084021176686885646320573268, 11.61430812631216105871510866690, 11.81113829924476911159363322084
