| L(s) = 1 | + 4·2-s − 8·3-s + 12·4-s − 5·5-s − 32·6-s − 14·7-s + 32·8-s − 6·9-s − 20·10-s + 22·11-s − 96·12-s − 56·14-s + 40·15-s + 80·16-s − 74·17-s − 24·18-s − 41·19-s − 60·20-s + 112·21-s + 88·22-s + 105·23-s − 256·24-s + 61·25-s + 392·27-s − 168·28-s + 341·29-s + 160·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.53·3-s + 3/2·4-s − 0.447·5-s − 2.17·6-s − 0.755·7-s + 1.41·8-s − 2/9·9-s − 0.632·10-s + 0.603·11-s − 2.30·12-s − 1.06·14-s + 0.688·15-s + 5/4·16-s − 1.05·17-s − 0.314·18-s − 0.495·19-s − 0.670·20-s + 1.16·21-s + 0.852·22-s + 0.951·23-s − 2.17·24-s + 0.487·25-s + 2.79·27-s − 1.13·28-s + 2.18·29-s + 0.973·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + p T - 36 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 1614 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 74 T + 10026 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 41 T - 182 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 105 T + 26798 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 341 T + 77556 T^{2} - 341 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 81 T + 37550 T^{2} + 81 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 378 T + 135858 T^{2} + 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 672 T + 246062 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 163 T + 163026 T^{2} + 163 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 61 T + 79694 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 168884 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 484 T + 240198 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 108 T + 157614 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 518 T + 667438 T^{2} - 518 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1336 T + 993710 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1335 T + 1216284 T^{2} + 1335 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1431 T + 1413558 T^{2} + 1431 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1747 T + 1801074 T^{2} + 1747 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 193 T + 1061244 T^{2} - 193 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1595 T + 1815772 T^{2} + 1595 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.476088552085037745573926499430, −8.186270974943796645155878109223, −7.08762417797894855740810898998, −6.95220269372537596223137354935, −6.85839810251648958785461763935, −6.34405439342038132053796092821, −6.17478075751049121811962919268, −5.57729618711272406085822116873, −5.17508233011128265670205156571, −5.06742963333999857263488454760, −4.47982527325111581552689113585, −4.22149319887176056958981587136, −3.40869575844716598984813357735, −3.31324294067272613993237208842, −2.74046101809348913793398766948, −2.28878782595099554496952072558, −1.44061258118458080062693752912, −0.912627117124973737855348940271, 0, 0,
0.912627117124973737855348940271, 1.44061258118458080062693752912, 2.28878782595099554496952072558, 2.74046101809348913793398766948, 3.31324294067272613993237208842, 3.40869575844716598984813357735, 4.22149319887176056958981587136, 4.47982527325111581552689113585, 5.06742963333999857263488454760, 5.17508233011128265670205156571, 5.57729618711272406085822116873, 6.17478075751049121811962919268, 6.34405439342038132053796092821, 6.85839810251648958785461763935, 6.95220269372537596223137354935, 7.08762417797894855740810898998, 8.186270974943796645155878109223, 8.476088552085037745573926499430
