| L(s) = 1 | − 2-s − 7·4-s + 10·5-s + 9·7-s + 7·8-s − 10·10-s − 37·11-s − 112·13-s − 9·14-s − 7·16-s + 77·17-s + 35·19-s − 70·20-s + 37·22-s − 267·23-s + 75·25-s + 112·26-s − 63·28-s + 325·29-s − 12·31-s + 71·32-s − 77·34-s + 90·35-s − 638·37-s − 35·38-s + 70·40-s + 238·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.894·5-s + 0.485·7-s + 0.309·8-s − 0.316·10-s − 1.01·11-s − 2.38·13-s − 0.171·14-s − 0.109·16-s + 1.09·17-s + 0.422·19-s − 0.782·20-s + 0.358·22-s − 2.42·23-s + 3/5·25-s + 0.844·26-s − 0.425·28-s + 2.08·29-s − 0.0695·31-s + 0.392·32-s − 0.388·34-s + 0.434·35-s − 2.83·37-s − 0.149·38-s + 0.276·40-s + 0.906·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 698 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 37 T + 2798 T^{2} + 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 112 T + 7002 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 77 T + 6152 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 35 T + 8868 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 267 T + 37792 T^{2} + 267 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 325 T + 62636 T^{2} - 325 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 638 T + 5466 p T^{2} + 638 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 238 T + 2983 p T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 97 T + 98922 T^{2} + 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 901 T + 409598 T^{2} + 901 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 224 T + 227798 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 85 T + 392756 T^{2} + 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 247 T + 155508 T^{2} + 247 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 606 T + 414023 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 394 T + 654806 T^{2} - 394 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 811 T + 595758 T^{2} + 811 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 840 T + 1114826 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 387 T - 511958 T^{2} + 387 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1065 T + 874888 T^{2} + 1065 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1031 T + 1949508 T^{2} + 1031 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.18968925532739595652587963361, −10.12249416728025618057410008971, −9.666767985475558314655697679348, −9.640904232392103980889346750376, −8.560437077153402890418273090342, −8.460780380512245500372213877398, −7.896209725887110860179543142108, −7.50137362910994330014738642677, −6.90805832120492354225694555464, −6.32431035508479658024644789136, −5.58985824535337388471699439136, −5.23699256741440599580335964172, −4.66977911570974299413027372415, −4.55736913378278604035829951614, −3.34138618707949579766015942790, −2.80074063198763675994116546125, −2.06929188837547122907173197286, −1.46206459093805403087378164364, 0, 0,
1.46206459093805403087378164364, 2.06929188837547122907173197286, 2.80074063198763675994116546125, 3.34138618707949579766015942790, 4.55736913378278604035829951614, 4.66977911570974299413027372415, 5.23699256741440599580335964172, 5.58985824535337388471699439136, 6.32431035508479658024644789136, 6.90805832120492354225694555464, 7.50137362910994330014738642677, 7.896209725887110860179543142108, 8.460780380512245500372213877398, 8.560437077153402890418273090342, 9.640904232392103980889346750376, 9.666767985475558314655697679348, 10.12249416728025618057410008971, 10.18968925532739595652587963361
