| L(s) = 1 | + 2·2-s + 2·4-s + 3·9-s − 4·16-s + 6·18-s − 25-s + 2·29-s − 8·32-s + 6·36-s − 7·49-s − 2·50-s − 20·53-s + 4·58-s − 8·64-s − 14·98-s − 2·100-s − 40·106-s + 30·109-s + 16·113-s + 4·116-s − 21·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 9-s − 16-s + 1.41·18-s − 1/5·25-s + 0.371·29-s − 1.41·32-s + 36-s − 49-s − 0.282·50-s − 2.74·53-s + 0.525·58-s − 64-s − 1.41·98-s − 1/5·100-s − 3.88·106-s + 2.87·109-s + 1.50·113-s + 0.371·116-s − 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.301409224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301409224\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−11.16082934664422888507415664885, −10.42703357898353451171232561647, −9.877191879518267225385464097371, −9.375538905149461374318659678050, −8.749867315934444980801831391315, −7.969282821348508699416415622410, −7.43451974256591676697530133038, −6.68496359825772854651235963790, −6.31780104077392117354355047996, −5.59619941150472580186581344036, −4.80025950309901808481083600396, −4.47811276515376096082407721051, −3.64223900207711112639934693945, −2.96737343451265466950866403592, −1.80569769789110742560219528774,
1.80569769789110742560219528774, 2.96737343451265466950866403592, 3.64223900207711112639934693945, 4.47811276515376096082407721051, 4.80025950309901808481083600396, 5.59619941150472580186581344036, 6.31780104077392117354355047996, 6.68496359825772854651235963790, 7.43451974256591676697530133038, 7.969282821348508699416415622410, 8.749867315934444980801831391315, 9.375538905149461374318659678050, 9.877191879518267225385464097371, 10.42703357898353451171232561647, 11.16082934664422888507415664885
