| L(s) = 1 | − 4·9-s − 4·11-s − 16·23-s − 10·25-s + 12·29-s − 4·37-s − 12·43-s + 12·53-s − 24·67-s + 8·71-s − 24·79-s + 7·81-s + 16·99-s − 8·107-s + 36·109-s − 24·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 1.20·11-s − 3.33·23-s − 2·25-s + 2.22·29-s − 0.657·37-s − 1.82·43-s + 1.64·53-s − 2.93·67-s + 0.949·71-s − 2.70·79-s + 7/9·81-s + 1.60·99-s − 0.773·107-s + 3.44·109-s − 2.25·113-s − 0.909·121-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 − 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 144 T^{2} + p^{2} T^{4} \) |
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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
−9.038431681103424148294092035541, −8.796411830139782650515102904746, −8.332783511928667936434927338805, −8.121401070290854432438867330597, −7.66893629872196205128658087659, −7.52330076307777207232084780656, −6.62059559230917785167409095120, −6.36161183897516140287053002425, −5.82698469204577047203088655363, −5.71664717735909934406597119967, −5.15465608247286389502543564364, −4.70027223285448322345911122155, −3.91638252635517787482122209919, −3.90648630375322893589313542854, −2.98099328889607406564341366270, −2.65870680996730301452400797456, −2.12947067911120445145131286293, −1.52111127652180586649059427526, 0, 0,
1.52111127652180586649059427526, 2.12947067911120445145131286293, 2.65870680996730301452400797456, 2.98099328889607406564341366270, 3.90648630375322893589313542854, 3.91638252635517787482122209919, 4.70027223285448322345911122155, 5.15465608247286389502543564364, 5.71664717735909934406597119967, 5.82698469204577047203088655363, 6.36161183897516140287053002425, 6.62059559230917785167409095120, 7.52330076307777207232084780656, 7.66893629872196205128658087659, 8.121401070290854432438867330597, 8.332783511928667936434927338805, 8.796411830139782650515102904746, 9.038431681103424148294092035541
