| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 7·13-s + 16-s + 6·22-s + 3·23-s + 2·25-s + 7·26-s + 32-s − 17·37-s + 6·44-s + 3·46-s + 6·47-s − 4·49-s + 2·50-s + 7·52-s + 3·59-s − 5·61-s + 64-s − 9·71-s − 8·73-s − 17·74-s + 6·83-s + 6·88-s + 3·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 1.94·13-s + 1/4·16-s + 1.27·22-s + 0.625·23-s + 2/5·25-s + 1.37·26-s + 0.176·32-s − 2.79·37-s + 0.904·44-s + 0.442·46-s + 0.875·47-s − 4/7·49-s + 0.282·50-s + 0.970·52-s + 0.390·59-s − 0.640·61-s + 1/8·64-s − 1.06·71-s − 0.936·73-s − 1.97·74-s + 0.658·83-s + 0.639·88-s + 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.393626347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.393626347\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.960865110208532572073026872103, −8.637295360903926283760638534109, −8.294906559543423058875466172457, −7.41336714477434261172621586277, −7.01614033249322101532911667771, −6.49670626739883749240087208333, −6.27421406868778568230548141843, −5.60303328263850849400202950398, −5.15280477815619461574815286605, −4.36158610026643344805249204285, −3.88580106778109219278090941820, −3.51160690888276001095495063565, −2.88930616146844710468768697881, −1.67387743585758454215397662536, −1.26022754237539906715805013491,
1.26022754237539906715805013491, 1.67387743585758454215397662536, 2.88930616146844710468768697881, 3.51160690888276001095495063565, 3.88580106778109219278090941820, 4.36158610026643344805249204285, 5.15280477815619461574815286605, 5.60303328263850849400202950398, 6.27421406868778568230548141843, 6.49670626739883749240087208333, 7.01614033249322101532911667771, 7.41336714477434261172621586277, 8.294906559543423058875466172457, 8.637295360903926283760638534109, 8.960865110208532572073026872103
