| L(s) = 1 | − 9-s − 6·11-s + 2·23-s + 3·25-s + 4·29-s + 10·37-s − 22·53-s − 6·67-s − 16·71-s − 6·79-s − 8·81-s + 6·99-s − 10·107-s + 10·109-s + 12·113-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.80·11-s + 0.417·23-s + 3/5·25-s + 0.742·29-s + 1.64·37-s − 3.02·53-s − 0.733·67-s − 1.89·71-s − 0.675·79-s − 8/9·81-s + 0.603·99-s − 0.966·107-s + 0.957·109-s + 1.12·113-s + 9/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 87 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
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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
−8.557010093045348102536618449089, −8.099867485039271274598522882843, −7.64123722092648744015596754170, −7.36787716102719307792514769387, −6.68654085368964267215353529921, −6.05265711664714168724471413685, −5.83999628326159028672234249784, −5.04673007165552684072355737806, −4.76581470016109825254177688586, −4.26391028954185497662374033879, −3.19767813668070076465998455534, −2.93426280655909862612248394903, −2.34499667772288137351513991014, −1.29067463708373424838427239177, 0,
1.29067463708373424838427239177, 2.34499667772288137351513991014, 2.93426280655909862612248394903, 3.19767813668070076465998455534, 4.26391028954185497662374033879, 4.76581470016109825254177688586, 5.04673007165552684072355737806, 5.83999628326159028672234249784, 6.05265711664714168724471413685, 6.68654085368964267215353529921, 7.36787716102719307792514769387, 7.64123722092648744015596754170, 8.099867485039271274598522882843, 8.557010093045348102536618449089
