| L(s) = 1 | − 4-s + 4·13-s − 3·16-s − 4·19-s − 2·25-s + 4·31-s + 8·37-s + 12·43-s − 14·49-s − 4·52-s + 4·61-s + 7·64-s − 16·67-s + 12·73-s + 4·76-s + 11·79-s + 12·97-s + 2·100-s + 8·103-s − 8·109-s + 2·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.10·13-s − 3/4·16-s − 0.917·19-s − 2/5·25-s + 0.718·31-s + 1.31·37-s + 1.82·43-s − 2·49-s − 0.554·52-s + 0.512·61-s + 7/8·64-s − 1.95·67-s + 1.40·73-s + 0.458·76-s + 1.23·79-s + 1.21·97-s + 1/5·100-s + 0.788·103-s − 0.766·109-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1158219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1158219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.537550449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537550449\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| 181 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 22 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.086594777047052415469651733751, −7.69047950587939609701088851107, −7.24821828681392525508309058814, −6.54265702612165485293784921493, −6.24006749930437913920641783753, −6.06040372123502111952086396956, −5.33077794941601381876287434396, −4.77260494945148412990214613463, −4.44355048222964503162772221272, −3.93290404033922012927374106496, −3.52187872353031073787961923583, −2.73468190582775845137920912454, −2.25094355896087869830979718119, −1.45422439408894145049745989647, −0.58359369120134607365634158410,
0.58359369120134607365634158410, 1.45422439408894145049745989647, 2.25094355896087869830979718119, 2.73468190582775845137920912454, 3.52187872353031073787961923583, 3.93290404033922012927374106496, 4.44355048222964503162772221272, 4.77260494945148412990214613463, 5.33077794941601381876287434396, 6.06040372123502111952086396956, 6.24006749930437913920641783753, 6.54265702612165485293784921493, 7.24821828681392525508309058814, 7.69047950587939609701088851107, 8.086594777047052415469651733751
