| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s + 6·11-s + 4·15-s − 10·17-s + 10·19-s − 4·21-s − 2·23-s − 25-s + 4·27-s − 10·29-s + 12·33-s − 4·35-s + 6·45-s − 10·47-s + 2·49-s − 20·51-s + 12·53-s + 12·55-s + 20·57-s + 10·59-s + 2·61-s − 6·63-s − 4·69-s + 10·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 1.80·11-s + 1.03·15-s − 2.42·17-s + 2.29·19-s − 0.872·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s − 1.85·29-s + 2.08·33-s − 0.676·35-s + 0.894·45-s − 1.45·47-s + 2/7·49-s − 2.80·51-s + 1.64·53-s + 1.61·55-s + 2.64·57-s + 1.30·59-s + 0.256·61-s − 0.755·63-s − 0.481·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.572111675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.572111675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.306202233089326937573211172438, −9.299818163094991104988386363056, −8.759200644373728656322773109326, −8.329854821010345677633876227303, −7.904813149223611782362745158290, −7.35840381121081107194852037529, −6.87871616872090958716222172185, −6.85887510045200398964861731518, −6.24062350111043808535716485891, −6.02655197494302531592264302422, −5.24292682450979807576732125990, −5.08353828982632020891760099111, −4.25821716851204574824643313665, −3.81635459738786532186489949853, −3.66450658614603189825287911964, −3.11355236406740113259478675107, −2.35470729885386349287976477235, −2.07043665250204080716950930375, −1.57065651195497396738607667842, −0.72391783017636444857978596323,
0.72391783017636444857978596323, 1.57065651195497396738607667842, 2.07043665250204080716950930375, 2.35470729885386349287976477235, 3.11355236406740113259478675107, 3.66450658614603189825287911964, 3.81635459738786532186489949853, 4.25821716851204574824643313665, 5.08353828982632020891760099111, 5.24292682450979807576732125990, 6.02655197494302531592264302422, 6.24062350111043808535716485891, 6.85887510045200398964861731518, 6.87871616872090958716222172185, 7.35840381121081107194852037529, 7.904813149223611782362745158290, 8.329854821010345677633876227303, 8.759200644373728656322773109326, 9.299818163094991104988386363056, 9.306202233089326937573211172438
