| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 6·11-s + 4·15-s + 2·23-s − 25-s − 4·27-s − 8·31-s + 12·33-s + 4·37-s + 2·45-s + 2·47-s − 6·49-s + 4·53-s + 12·55-s + 16·59-s + 10·67-s + 4·69-s + 4·71-s − 2·75-s − 11·81-s + 32·89-s − 16·93-s − 12·97-s + 6·99-s − 22·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.80·11-s + 1.03·15-s + 0.417·23-s − 1/5·25-s − 0.769·27-s − 1.43·31-s + 2.08·33-s + 0.657·37-s + 0.298·45-s + 0.291·47-s − 6/7·49-s + 0.549·53-s + 1.61·55-s + 2.08·59-s + 1.22·67-s + 0.481·69-s + 0.474·71-s − 0.230·75-s − 1.22·81-s + 3.39·89-s − 1.65·93-s − 1.21·97-s + 0.603·99-s − 2.16·103-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
| \(L(1)\) |
\(\approx\) |
\(4.705849741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.705849741\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.992702249212472657533728301942, −7.35274570434748825354418995695, −6.96902331401254767443878249953, −6.63266076744712491907513518911, −6.16950901814084364897872247632, −5.60271339174011198954284276409, −5.41154997542091708326734134987, −4.63043939344072261016224231478, −4.11498963178225490542874836768, −3.60182667398516046267030120898, −3.42260147080236537370067647492, −2.57426604435551614905342585177, −2.10457637657166204437880532536, −1.67785896414465448623758569529, −0.870780218679902862613190774276,
0.870780218679902862613190774276, 1.67785896414465448623758569529, 2.10457637657166204437880532536, 2.57426604435551614905342585177, 3.42260147080236537370067647492, 3.60182667398516046267030120898, 4.11498963178225490542874836768, 4.63043939344072261016224231478, 5.41154997542091708326734134987, 5.60271339174011198954284276409, 6.16950901814084364897872247632, 6.63266076744712491907513518911, 6.96902331401254767443878249953, 7.35274570434748825354418995695, 7.992702249212472657533728301942
