| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s − 15·5-s − 24·6-s + 9·7-s − 32·8-s + 27·9-s + 60·10-s − 9·11-s + 72·12-s + 48·13-s − 36·14-s − 90·15-s + 80·16-s + 111·17-s − 108·18-s − 180·20-s + 54·21-s + 36·22-s − 138·23-s − 192·24-s − 3·25-s − 192·26-s + 108·27-s + 108·28-s − 390·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s + 0.485·7-s − 1.41·8-s + 9-s + 1.89·10-s − 0.246·11-s + 1.73·12-s + 1.02·13-s − 0.687·14-s − 1.54·15-s + 5/4·16-s + 1.58·17-s − 1.41·18-s − 2.01·20-s + 0.561·21-s + 0.348·22-s − 1.25·23-s − 1.63·24-s − 0.0239·25-s − 1.44·26-s + 0.769·27-s + 0.728·28-s − 2.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 19 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 3 p T + 228 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 2 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 9 T + 2604 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 111 T + 9072 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 p T + 21270 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 390 T + 83986 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 192 T + 23726 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 198 T + 144826 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 167 T + 7530 T^{2} + 167 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 93 T + 65124 T^{2} - 93 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 162 T + 233890 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 108 T + 131974 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 213 T + 306848 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 204 T + 510518 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1752 T + 1438126 T^{2} - 1752 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1417 T + 1273668 T^{2} + 1417 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1092 T + 732062 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 270 T + 733302 T^{2} + 270 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 606 T + 1025674 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 612 T + 1366850 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.406050298170214614428854105502, −8.316026923950485432714553139742, −7.78654091329289451347573238852, −7.60669638773358294684183115971, −7.27219021523867544038231941295, −7.02943810426142439340601831838, −6.13543732703752158715179978675, −6.01992048008703750708963931976, −5.42796834445526076215166904239, −4.90144481820543894442052043424, −4.15769524906256650324045840471, −3.82900181217596025825763248838, −3.39700238435379621858916279861, −3.29810630509676954485769148805, −2.31719116656515802079999682254, −2.09856282336838448588887092061, −1.26404026588686273557587599389, −1.19306389200072773520898345287, 0, 0,
1.19306389200072773520898345287, 1.26404026588686273557587599389, 2.09856282336838448588887092061, 2.31719116656515802079999682254, 3.29810630509676954485769148805, 3.39700238435379621858916279861, 3.82900181217596025825763248838, 4.15769524906256650324045840471, 4.90144481820543894442052043424, 5.42796834445526076215166904239, 6.01992048008703750708963931976, 6.13543732703752158715179978675, 7.02943810426142439340601831838, 7.27219021523867544038231941295, 7.60669638773358294684183115971, 7.78654091329289451347573238852, 8.316026923950485432714553139742, 8.406050298170214614428854105502
