| L(s) = 1 | − 3-s − 5-s − 2·9-s + 15-s − 4·25-s + 5·27-s + 7·31-s − 10·37-s + 2·45-s − 12·49-s + 5·53-s + 23·67-s − 5·71-s + 4·75-s + 81-s − 15·89-s − 7·93-s + 2·97-s + 7·103-s + 10·111-s − 11·121-s + 9·125-s + 127-s + 131-s − 5·135-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.258·15-s − 4/5·25-s + 0.962·27-s + 1.25·31-s − 1.64·37-s + 0.298·45-s − 1.71·49-s + 0.686·53-s + 2.80·67-s − 0.593·71-s + 0.461·75-s + 1/9·81-s − 1.58·89-s − 0.725·93-s + 0.203·97-s + 0.689·103-s + 0.949·111-s − 121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s − 0.430·135-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.67520839942905594114393746180, −7.04090172477884231390057022946, −6.76283727981711591414679478413, −6.37587706260337186325792624655, −5.85018165883120491676880071524, −5.41545148154031037180685890793, −5.10825549579191978930338108425, −4.51229760191079839160566714753, −4.11773678721840727067461631834, −3.41665505325270525353388505640, −3.13779755099539190022857411522, −2.39111425625291131729749674026, −1.78525573442604492698123330272, −0.852582563367677790063020673065, 0,
0.852582563367677790063020673065, 1.78525573442604492698123330272, 2.39111425625291131729749674026, 3.13779755099539190022857411522, 3.41665505325270525353388505640, 4.11773678721840727067461631834, 4.51229760191079839160566714753, 5.10825549579191978930338108425, 5.41545148154031037180685890793, 5.85018165883120491676880071524, 6.37587706260337186325792624655, 6.76283727981711591414679478413, 7.04090172477884231390057022946, 7.67520839942905594114393746180
