| L(s) = 1 | + 4-s + 6·5-s + 16-s + 6·20-s + 17·25-s − 8·31-s − 2·37-s + 24·47-s + 2·49-s + 12·53-s + 64-s − 8·67-s + 24·71-s + 6·80-s + 6·89-s + 4·97-s + 17·100-s − 8·103-s + 30·113-s − 11·121-s − 8·124-s + 18·125-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.68·5-s + 1/4·16-s + 1.34·20-s + 17/5·25-s − 1.43·31-s − 0.328·37-s + 3.50·47-s + 2/7·49-s + 1.64·53-s + 1/8·64-s − 0.977·67-s + 2.84·71-s + 0.670·80-s + 0.635·89-s + 0.406·97-s + 1.69·100-s − 0.788·103-s + 2.82·113-s − 121-s − 0.718·124-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.315578076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.315578076\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.24922326860214571213476798386, −7.24345462277249174674167705434, −6.53027619996145516090571249912, −6.18735538198950166736525445007, −5.89516609795681791126651546534, −5.49090131952205180218546644094, −5.28450148258092108190969844265, −4.77055228819099980126958913880, −3.88992644556742170401529174780, −3.69589098584017718744945020548, −2.76020437849783312932298367516, −2.30016225881827778465495235815, −2.18530917563711206992052355334, −1.51756437179293203913385442880, −0.875045304568839828134702749689,
0.875045304568839828134702749689, 1.51756437179293203913385442880, 2.18530917563711206992052355334, 2.30016225881827778465495235815, 2.76020437849783312932298367516, 3.69589098584017718744945020548, 3.88992644556742170401529174780, 4.77055228819099980126958913880, 5.28450148258092108190969844265, 5.49090131952205180218546644094, 5.89516609795681791126651546534, 6.18735538198950166736525445007, 6.53027619996145516090571249912, 7.24345462277249174674167705434, 7.24922326860214571213476798386
