| L(s) = 1 | − 4-s + 4·9-s − 6·11-s − 3·16-s − 2·19-s − 5·25-s + 12·29-s − 8·31-s − 4·36-s + 6·41-s + 6·44-s − 4·49-s + 12·59-s + 10·61-s + 7·64-s + 12·71-s + 2·76-s − 14·79-s + 7·81-s + 5·89-s − 24·99-s + 5·100-s − 24·101-s + 4·109-s − 12·116-s + 14·121-s + 8·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 4/3·9-s − 1.80·11-s − 3/4·16-s − 0.458·19-s − 25-s + 2.22·29-s − 1.43·31-s − 2/3·36-s + 0.937·41-s + 0.904·44-s − 4/7·49-s + 1.56·59-s + 1.28·61-s + 7/8·64-s + 1.42·71-s + 0.229·76-s − 1.57·79-s + 7/9·81-s + 0.529·89-s − 2.41·99-s + 1/2·100-s − 2.38·101-s + 0.383·109-s − 1.11·116-s + 1.27·121-s + 0.718·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6102056362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6102056362\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−13.05928479743213572873466761523, −12.84199855774120840044371240946, −12.15280748415887504594914016442, −11.22817843905631117413333233882, −10.65647438130879238626828408512, −9.994078764977635287414589724651, −9.652563093560623762919147238089, −8.630533751891957715111021430614, −8.075123975720972663658268086685, −7.31944436042866177881822636165, −6.64487784227063850738118989129, −5.50489121824126282229787546052, −4.76184095572488922680179662262, −3.97526270023131261304188897360, −2.41736965179828481146939160329,
2.41736965179828481146939160329, 3.97526270023131261304188897360, 4.76184095572488922680179662262, 5.50489121824126282229787546052, 6.64487784227063850738118989129, 7.31944436042866177881822636165, 8.075123975720972663658268086685, 8.630533751891957715111021430614, 9.652563093560623762919147238089, 9.994078764977635287414589724651, 10.65647438130879238626828408512, 11.22817843905631117413333233882, 12.15280748415887504594914016442, 12.84199855774120840044371240946, 13.05928479743213572873466761523
