| L(s) = 1 | + 4·2-s + 12·4-s + 14·5-s + 4·7-s + 32·8-s + 56·10-s + 44·11-s − 32·13-s + 16·14-s + 80·16-s + 114·17-s − 12·19-s + 168·20-s + 176·22-s + 166·23-s − 4·25-s − 128·26-s + 48·28-s + 114·29-s + 32·31-s + 192·32-s + 456·34-s + 56·35-s − 74·37-s − 48·38-s + 448·40-s + 268·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.25·5-s + 0.215·7-s + 1.41·8-s + 1.77·10-s + 1.20·11-s − 0.682·13-s + 0.305·14-s + 5/4·16-s + 1.62·17-s − 0.144·19-s + 1.87·20-s + 1.70·22-s + 1.50·23-s − 0.0319·25-s − 0.965·26-s + 0.323·28-s + 0.729·29-s + 0.185·31-s + 1.06·32-s + 2.30·34-s + 0.270·35-s − 0.328·37-s − 0.204·38-s + 1.77·40-s + 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.93950607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.93950607\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 14 T + 8 p^{2} T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 514 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 186 p T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 32 T + 4254 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 114 T + 11216 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12 T + 11598 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 166 T + 30332 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 114 T + 46208 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 32 T + 14782 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 268 T + 155622 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480 T + 165750 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 568 T + 203118 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 440 T + 322878 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 746 T + 503412 T^{2} - 746 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 220 T + 457438 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 300 T + 383082 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 474878 T^{2} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1020 T + 884970 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 780 T + 1071254 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 820 T + 1298958 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 90 T + 1234544 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 128 T - 1260458 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.24198108086762090435194124366, −10.04451591909922591428735554786, −9.425048352428359237615934243837, −9.393015094476374624289719332563, −8.357847349857883293285931915943, −8.324035003236834638321448991394, −7.32889969899286550678123029264, −7.18993705538920743171791706820, −6.56408895730927025375355885359, −6.30139640316624844280764957915, −5.59320378812772118943819605914, −5.43753948536666633358392099551, −4.92363817793157402414684746004, −4.43126202560322281036547302454, −3.63164361862187837666731138807, −3.44190805218523520705013029437, −2.52015978707112369201692177883, −2.23919929012018438459231463937, −1.33824865455340953180653009106, −0.955203896489273506374242700011,
0.955203896489273506374242700011, 1.33824865455340953180653009106, 2.23919929012018438459231463937, 2.52015978707112369201692177883, 3.44190805218523520705013029437, 3.63164361862187837666731138807, 4.43126202560322281036547302454, 4.92363817793157402414684746004, 5.43753948536666633358392099551, 5.59320378812772118943819605914, 6.30139640316624844280764957915, 6.56408895730927025375355885359, 7.18993705538920743171791706820, 7.32889969899286550678123029264, 8.324035003236834638321448991394, 8.357847349857883293285931915943, 9.393015094476374624289719332563, 9.425048352428359237615934243837, 10.04451591909922591428735554786, 10.24198108086762090435194124366
