| L(s) = 1 | + 3·4-s + 2·5-s + 2·9-s − 4·11-s + 5·16-s + 8·19-s + 6·20-s − 25-s − 7·31-s + 6·36-s − 12·44-s + 4·45-s − 2·49-s − 8·55-s + 8·59-s + 8·61-s + 3·64-s − 8·71-s + 24·76-s − 16·79-s + 10·80-s − 5·81-s − 4·89-s + 16·95-s − 8·99-s − 3·100-s − 16·101-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.894·5-s + 2/3·9-s − 1.20·11-s + 5/4·16-s + 1.83·19-s + 1.34·20-s − 1/5·25-s − 1.25·31-s + 36-s − 1.80·44-s + 0.596·45-s − 2/7·49-s − 1.07·55-s + 1.04·59-s + 1.02·61-s + 3/8·64-s − 0.949·71-s + 2.75·76-s − 1.80·79-s + 1.11·80-s − 5/9·81-s − 0.423·89-s + 1.64·95-s − 0.804·99-s − 0.299·100-s − 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93775 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93775 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.588132054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.588132054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.726597738252571816248750693773, −9.335222371110420346168735944662, −8.593038044375713312254455647003, −7.926108447559249870491031456688, −7.49208559131459552698013567851, −7.10715004057764079702405178505, −6.73557051920693071637965066956, −5.91249013709413575902171719023, −5.55938863144622067686870599273, −5.20523044814096307458678241351, −4.24882783845731345413922216699, −3.33167730593220996016771650629, −2.77262963906123951720892764968, −2.08921498349059695430709819890, −1.39963279110069445579720114227,
1.39963279110069445579720114227, 2.08921498349059695430709819890, 2.77262963906123951720892764968, 3.33167730593220996016771650629, 4.24882783845731345413922216699, 5.20523044814096307458678241351, 5.55938863144622067686870599273, 5.91249013709413575902171719023, 6.73557051920693071637965066956, 7.10715004057764079702405178505, 7.49208559131459552698013567851, 7.926108447559249870491031456688, 8.593038044375713312254455647003, 9.335222371110420346168735944662, 9.726597738252571816248750693773
