| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·15-s + 4·17-s + 2·25-s + 27-s + 12·37-s − 12·41-s − 8·43-s − 4·45-s − 7·49-s + 4·51-s − 8·59-s + 8·67-s + 2·75-s + 16·79-s + 81-s + 8·83-s − 16·85-s − 12·89-s − 36·101-s − 4·109-s + 12·111-s − 6·121-s − 12·123-s + 28·125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.03·15-s + 0.970·17-s + 2/5·25-s + 0.192·27-s + 1.97·37-s − 1.87·41-s − 1.21·43-s − 0.596·45-s − 49-s + 0.560·51-s − 1.04·59-s + 0.977·67-s + 0.230·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 1.73·85-s − 1.27·89-s − 3.58·101-s − 0.383·109-s + 1.13·111-s − 0.545·121-s − 1.08·123-s + 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.173282349834764667604075089133, −8.131315776481593009713606089546, −7.82263580546813636934054242173, −7.26958585488936859387169650645, −6.78395093605661555810316280628, −6.30966811163732211212778691110, −5.58667709375314371155679753614, −5.01357758335939619070346679818, −4.44811958782670156328485077225, −3.97597569667552498946447381576, −3.42103471864385914094720232253, −3.14826068230388029805668446658, −2.20947719434785265756306128762, −1.24950065754749508798543005172, 0,
1.24950065754749508798543005172, 2.20947719434785265756306128762, 3.14826068230388029805668446658, 3.42103471864385914094720232253, 3.97597569667552498946447381576, 4.44811958782670156328485077225, 5.01357758335939619070346679818, 5.58667709375314371155679753614, 6.30966811163732211212778691110, 6.78395093605661555810316280628, 7.26958585488936859387169650645, 7.82263580546813636934054242173, 8.131315776481593009713606089546, 8.173282349834764667604075089133
