| L(s) = 1 | + 2·3-s + 4·5-s + 44·7-s − 51·9-s − 40·11-s + 20·13-s + 8·15-s + 46·17-s − 38·19-s + 88·21-s − 220·23-s + 134·25-s − 158·27-s + 84·29-s − 84·31-s − 80·33-s + 176·35-s − 304·37-s + 40·39-s + 168·41-s + 372·43-s − 204·45-s + 584·47-s + 859·49-s + 92·51-s − 484·53-s − 160·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.357·5-s + 2.37·7-s − 1.88·9-s − 1.09·11-s + 0.426·13-s + 0.137·15-s + 0.656·17-s − 0.458·19-s + 0.914·21-s − 1.99·23-s + 1.07·25-s − 1.12·27-s + 0.537·29-s − 0.486·31-s − 0.422·33-s + 0.849·35-s − 1.35·37-s + 0.164·39-s + 0.639·41-s + 1.31·43-s − 0.675·45-s + 1.81·47-s + 2.50·49-s + 0.252·51-s − 1.25·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.298812268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.298812268\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 - 4 T - 118 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 44 T + 1077 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 40 T + 2690 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 20 T + 3657 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 46 T + 259 p T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 220 T + 1483 p T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 84 T + 16969 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 84 T + 31214 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 304 T + 121062 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 168 T + 107698 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 372 T + 193238 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 584 T + 239342 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 484 T + 255041 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1074 T + 697639 T^{2} - 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 88 T + 437670 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1430 T + 1039839 T^{2} + 1430 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1848 T + 1569226 T^{2} - 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1294 T + 1190691 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 832 T + 593322 T^{2} - 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 528 T + 459970 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1844 T + 1750754 T^{2} - 1844 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 364 T + 1606998 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.197398166816076848853812137461, −9.143643637990003596728039989914, −8.658213100207168847929668342520, −8.204808452851937649675489846950, −7.997076187135639665498129466273, −7.86015310129791423137485735319, −7.31264805971668257374168860701, −6.63558425043175627857333775693, −5.92868618536553408998649041449, −5.78175010426549687860910451416, −5.23630052727194331490437794878, −5.13564065508851172415181808178, −4.39091155672215984628548418318, −4.00671011750871063449759003133, −3.28195956610430084038990752289, −2.74552881288828589826461210494, −2.13161113561608785090475217708, −2.01527396157207334617864057038, −1.13767762486003359729916879992, −0.42794692993666081577960750222,
0.42794692993666081577960750222, 1.13767762486003359729916879992, 2.01527396157207334617864057038, 2.13161113561608785090475217708, 2.74552881288828589826461210494, 3.28195956610430084038990752289, 4.00671011750871063449759003133, 4.39091155672215984628548418318, 5.13564065508851172415181808178, 5.23630052727194331490437794878, 5.78175010426549687860910451416, 5.92868618536553408998649041449, 6.63558425043175627857333775693, 7.31264805971668257374168860701, 7.86015310129791423137485735319, 7.997076187135639665498129466273, 8.204808452851937649675489846950, 8.658213100207168847929668342520, 9.143643637990003596728039989914, 9.197398166816076848853812137461
