| L(s) = 1 | − 6·9-s + 6·11-s − 6·17-s + 2·19-s − 9·25-s + 22·43-s − 13·49-s + 12·59-s − 20·67-s + 10·73-s + 27·81-s + 12·89-s − 24·97-s − 36·99-s + 20·107-s + 24·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 36·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·9-s + 1.80·11-s − 1.45·17-s + 0.458·19-s − 9/5·25-s + 3.35·43-s − 1.85·49-s + 1.56·59-s − 2.44·67-s + 1.17·73-s + 3·81-s + 1.27·89-s − 2.43·97-s − 3.61·99-s + 1.93·107-s + 2.25·113-s + 5/11·121-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 + 2.91·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.265849533784556159385578271594, −7.59019329531238695962577712799, −7.22663142743355843577020555382, −6.57705024982005416568601729042, −6.23231971111290599279730913097, −5.71435981136868716408885186688, −5.69887754666040376701153368082, −4.57069618429307251486414062571, −4.47863079866238157165170735095, −3.55029150041215641334184982487, −3.47456597646201355004490006149, −2.41972145972916601985658200179, −2.19568966813265176024980758112, −1.10874592215816899307332244272, 0,
1.10874592215816899307332244272, 2.19568966813265176024980758112, 2.41972145972916601985658200179, 3.47456597646201355004490006149, 3.55029150041215641334184982487, 4.47863079866238157165170735095, 4.57069618429307251486414062571, 5.69887754666040376701153368082, 5.71435981136868716408885186688, 6.23231971111290599279730913097, 6.57705024982005416568601729042, 7.22663142743355843577020555382, 7.59019329531238695962577712799, 8.265849533784556159385578271594
