| L(s) = 1 | + 3-s + 4-s − 2·7-s + 9-s + 12-s + 13-s + 16-s + 13·19-s − 2·21-s − 25-s + 27-s − 2·28-s + 4·31-s + 36-s − 8·37-s + 39-s + 43-s + 48-s − 2·49-s + 52-s + 13·57-s − 11·61-s − 2·63-s + 64-s + 28·67-s − 23·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 2.98·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 0.377·28-s + 0.718·31-s + 1/6·36-s − 1.31·37-s + 0.160·39-s + 0.152·43-s + 0.144·48-s − 2/7·49-s + 0.138·52-s + 1.72·57-s − 1.40·61-s − 0.251·63-s + 1/8·64-s + 3.42·67-s − 2.69·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465579396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465579396\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 2221 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 55 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−9.034683990093103938303216776739, −8.524036749430120218159616534725, −7.963395776349138708339080651868, −7.42631693582098025205158412453, −7.28684295346162243567408917035, −6.57231436670467153726536323715, −6.23765144919291731378633030953, −5.43389502981139771785502936856, −5.23275963224592600707487996187, −4.39832809117273558653331987491, −3.59130653981882330643180536895, −3.23718394798078720008693759065, −2.81262337445221762304378522922, −1.85797050018321080471027347098, −0.991769375036072643266844467989,
0.991769375036072643266844467989, 1.85797050018321080471027347098, 2.81262337445221762304378522922, 3.23718394798078720008693759065, 3.59130653981882330643180536895, 4.39832809117273558653331987491, 5.23275963224592600707487996187, 5.43389502981139771785502936856, 6.23765144919291731378633030953, 6.57231436670467153726536323715, 7.28684295346162243567408917035, 7.42631693582098025205158412453, 7.963395776349138708339080651868, 8.524036749430120218159616534725, 9.034683990093103938303216776739
