| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−9.128769234310573620005228056931, −8.917893205469915773158969755520, −8.485242741495349942554915888680, −8.083738722835255727420789698462, −7.33711514859523866606934098749, −6.75128826479355224509431381338, −6.56410156099193031788154077670, −6.41848853858587837272007550388, −5.79728242906429228223159347915, −5.55413727175344320027893379523, −4.93072113533786590667549632860, −4.55136195158787050176654012662, −3.49115604823327394888079237802, −3.28167449050424977321876635221, −3.02263423017133149104774076181, −2.71232307156100525535306455302, −1.45031275962286229185646455701, −1.04993588526087211607630818539, 0, 0,
1.04993588526087211607630818539, 1.45031275962286229185646455701, 2.71232307156100525535306455302, 3.02263423017133149104774076181, 3.28167449050424977321876635221, 3.49115604823327394888079237802, 4.55136195158787050176654012662, 4.93072113533786590667549632860, 5.55413727175344320027893379523, 5.79728242906429228223159347915, 6.41848853858587837272007550388, 6.56410156099193031788154077670, 6.75128826479355224509431381338, 7.33711514859523866606934098749, 8.083738722835255727420789698462, 8.485242741495349942554915888680, 8.917893205469915773158969755520, 9.128769234310573620005228056931
