| L(s) = 1 | + 2-s − 5-s − 2·7-s − 8-s − 10-s − 6·11-s + 4·13-s − 2·14-s − 16-s − 12·17-s − 8·19-s − 6·22-s + 4·26-s + 6·29-s + 4·31-s − 12·34-s + 2·35-s + 16·37-s − 8·38-s + 40-s − 8·43-s + 7·49-s − 12·53-s + 6·55-s + 2·56-s + 6·58-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 1.10·13-s − 0.534·14-s − 1/4·16-s − 2.91·17-s − 1.83·19-s − 1.27·22-s + 0.784·26-s + 1.11·29-s + 0.718·31-s − 2.05·34-s + 0.338·35-s + 2.63·37-s − 1.29·38-s + 0.158·40-s − 1.21·43-s + 49-s − 1.64·53-s + 0.809·55-s + 0.267·56-s + 0.787·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6723646700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6723646700\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.47971562234997681470438693717, −10.34589126991547620014561408667, −9.612177295253383939311759530411, −8.947470716530354110477846448794, −8.864964420080332731356074613770, −8.304652717885101175682476443910, −8.042113016837210658277325359435, −7.44881700850522627078526022028, −6.80134109538508459079806331620, −6.43996645843552131653350404093, −6.03857788877013046871244035435, −5.87517717497737689349266025337, −4.66655724417253610145915158142, −4.61645995469969852790622004617, −4.38993071773018760611484621593, −3.61171609615017146047630509246, −2.75808607643885212279123698217, −2.72105084104399368851911476777, −1.87107147946259653933931644430, −0.34198757622228646662232513751,
0.34198757622228646662232513751, 1.87107147946259653933931644430, 2.72105084104399368851911476777, 2.75808607643885212279123698217, 3.61171609615017146047630509246, 4.38993071773018760611484621593, 4.61645995469969852790622004617, 4.66655724417253610145915158142, 5.87517717497737689349266025337, 6.03857788877013046871244035435, 6.43996645843552131653350404093, 6.80134109538508459079806331620, 7.44881700850522627078526022028, 8.042113016837210658277325359435, 8.304652717885101175682476443910, 8.864964420080332731356074613770, 8.947470716530354110477846448794, 9.612177295253383939311759530411, 10.34589126991547620014561408667, 10.47971562234997681470438693717
