| L(s) = 1 | − 2-s + 2·7-s − 8-s + 6·11-s + 2·13-s − 2·14-s − 16-s − 2·17-s + 6·19-s − 6·22-s − 2·23-s − 2·26-s + 8·29-s + 6·32-s + 2·34-s + 6·37-s − 6·38-s + 12·43-s + 2·46-s − 14·47-s + 3·49-s + 8·53-s − 2·56-s − 8·58-s + 14·59-s + 12·61-s − 3·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.755·7-s − 0.353·8-s + 1.80·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s − 1.27·22-s − 0.417·23-s − 0.392·26-s + 1.48·29-s + 1.06·32-s + 0.342·34-s + 0.986·37-s − 0.973·38-s + 1.82·43-s + 0.294·46-s − 2.04·47-s + 3/7·49-s + 1.09·53-s − 0.267·56-s − 1.05·58-s + 1.82·59-s + 1.53·61-s − 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.145385444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145385444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24 T + 265 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 206 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.454702779453558603687495098362, −9.373305601241735506001126093248, −8.764442251009763260469150801085, −8.493121564571527519054078223001, −8.035371209682523015740046840102, −8.026108328558568297157779727521, −7.02542870717423248568970136929, −6.83007513077537615270558567408, −6.64305296539398744761913097100, −6.05318035986079963216235704218, −5.40939092430265726654078949254, −5.31607799296967689680114561723, −4.43774547754851425401365398106, −4.08574816439250349541748956655, −3.90131234534287410232399866259, −3.03268511837486717539072167919, −2.57144016263289016065000871477, −1.89105327039571605253244359212, −1.07921911161642979531968838501, −0.831712353303606640939264466187,
0.831712353303606640939264466187, 1.07921911161642979531968838501, 1.89105327039571605253244359212, 2.57144016263289016065000871477, 3.03268511837486717539072167919, 3.90131234534287410232399866259, 4.08574816439250349541748956655, 4.43774547754851425401365398106, 5.31607799296967689680114561723, 5.40939092430265726654078949254, 6.05318035986079963216235704218, 6.64305296539398744761913097100, 6.83007513077537615270558567408, 7.02542870717423248568970136929, 8.026108328558568297157779727521, 8.035371209682523015740046840102, 8.493121564571527519054078223001, 8.764442251009763260469150801085, 9.373305601241735506001126093248, 9.454702779453558603687495098362
