| L(s) = 1 | + 2·3-s − 4·5-s + 9-s − 8·15-s − 12·17-s + 6·25-s − 4·27-s + 4·37-s − 4·41-s + 4·43-s − 4·45-s + 8·47-s − 7·49-s − 24·51-s − 4·59-s + 12·67-s + 12·75-s + 24·79-s − 11·81-s + 20·83-s + 48·85-s + 4·89-s + 12·101-s − 12·109-s + 8·111-s + 10·121-s − 8·123-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1/3·9-s − 2.06·15-s − 2.91·17-s + 6/5·25-s − 0.769·27-s + 0.657·37-s − 0.624·41-s + 0.609·43-s − 0.596·45-s + 1.16·47-s − 49-s − 3.36·51-s − 0.520·59-s + 1.46·67-s + 1.38·75-s + 2.70·79-s − 1.22·81-s + 2.19·83-s + 5.20·85-s + 0.423·89-s + 1.19·101-s − 1.14·109-s + 0.759·111-s + 0.909·121-s − 0.721·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.115239002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115239002\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−8.505474113627186976215339908853, −8.071096866319721899408836574157, −7.894192387602802154760743051657, −7.33327971554839011819229001528, −6.86508117650847798131686774556, −6.48479861682674759579918431232, −5.89244822511649228149174122503, −4.95227331595035725359269006803, −4.59939540412711697889817652330, −4.08093481134776308724633229294, −3.69234875079526602690444014229, −3.20876670140457627419053348399, −2.35302370075336159766076853357, −2.08054414463286654291255684330, −0.50623413256191988188895227304,
0.50623413256191988188895227304, 2.08054414463286654291255684330, 2.35302370075336159766076853357, 3.20876670140457627419053348399, 3.69234875079526602690444014229, 4.08093481134776308724633229294, 4.59939540412711697889817652330, 4.95227331595035725359269006803, 5.89244822511649228149174122503, 6.48479861682674759579918431232, 6.86508117650847798131686774556, 7.33327971554839011819229001528, 7.894192387602802154760743051657, 8.071096866319721899408836574157, 8.505474113627186976215339908853
