| L(s) = 1 | + 2·2-s − 10·4-s − 32·8-s + 22·11-s + 64·13-s + 44·16-s + 32·17-s − 10·19-s + 44·22-s − 84·23-s + 128·26-s + 170·29-s − 258·31-s + 248·32-s + 64·34-s − 76·37-s − 20·38-s + 578·41-s − 380·43-s − 220·44-s − 168·46-s − 484·47-s − 638·49-s − 640·52-s − 544·53-s + 340·58-s + 706·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/4·4-s − 1.41·8-s + 0.603·11-s + 1.36·13-s + 0.687·16-s + 0.456·17-s − 0.120·19-s + 0.426·22-s − 0.761·23-s + 0.965·26-s + 1.08·29-s − 1.49·31-s + 1.37·32-s + 0.322·34-s − 0.337·37-s − 0.0853·38-s + 2.20·41-s − 1.34·43-s − 0.753·44-s − 0.538·46-s − 1.50·47-s − 1.86·49-s − 1.70·52-s − 1.40·53-s + 0.769·58-s + 1.55·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.811677253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.811677253\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 638 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 431 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 64 T + 5370 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 32 T + 674 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T - 129 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 84 T + 23026 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 170 T + 54275 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 258 T + 75791 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 76 T + 37038 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 578 T + 214451 T^{2} - 578 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 380 T + 122106 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 484 T + 265442 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 544 T + 341738 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 706 T + 533015 T^{2} - 706 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 668 T + 454926 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1452 T + 1120490 T^{2} - 1452 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 974 T + 837743 T^{2} - 974 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1184 T + 1103106 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 995246 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 444 T + 112858 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 513 T + p^{3} T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 668 T + 1356102 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.766908927215778153657656446375, −8.749461900020151695844185601355, −8.151204619052997956508844098877, −8.095950035754631556951029058420, −7.50315087238447962520955105473, −6.84260649794239645589614696946, −6.40749658649640642051362479877, −6.28090288521215783283842401046, −5.63528636348006035481728749733, −5.38966606169189828041120911317, −4.84644310076691336095778896523, −4.61170069994711412191698018094, −3.91776893986189184327818812622, −3.76713950206946813648973915527, −3.39229589053210417327257462861, −2.94064907923178983545511318476, −1.96553951951802745212655210226, −1.64072222919510160247815540038, −0.70997303690079513560626872202, −0.54761932907374758269699070969,
0.54761932907374758269699070969, 0.70997303690079513560626872202, 1.64072222919510160247815540038, 1.96553951951802745212655210226, 2.94064907923178983545511318476, 3.39229589053210417327257462861, 3.76713950206946813648973915527, 3.91776893986189184327818812622, 4.61170069994711412191698018094, 4.84644310076691336095778896523, 5.38966606169189828041120911317, 5.63528636348006035481728749733, 6.28090288521215783283842401046, 6.40749658649640642051362479877, 6.84260649794239645589614696946, 7.50315087238447962520955105473, 8.095950035754631556951029058420, 8.151204619052997956508844098877, 8.749461900020151695844185601355, 8.766908927215778153657656446375
