L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s − 10·5-s − 24·6-s − 32·8-s + 27·9-s + 40·10-s − 16·11-s + 72·12-s + 28·13-s − 60·15-s + 80·16-s + 32·17-s − 108·18-s − 100·19-s − 120·20-s + 64·22-s − 4·23-s − 192·24-s + 75·25-s − 112·26-s + 108·27-s + 148·29-s + 240·30-s − 132·31-s − 192·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 0.438·11-s + 1.73·12-s + 0.597·13-s − 1.03·15-s + 5/4·16-s + 0.456·17-s − 1.41·18-s − 1.20·19-s − 1.34·20-s + 0.620·22-s − 0.0362·23-s − 1.63·24-s + 3/5·25-s − 0.844·26-s + 0.769·27-s + 0.947·29-s + 1.46·30-s − 0.764·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
good | 11 | $D_{4}$ | \( 1 + 16 T + 2214 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 28 T + 4390 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 32 T + 10050 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 100 T + 14296 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 13680 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 148 T + 53454 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 132 T + 63216 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 460 T + 149206 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 284 T + 76398 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 192 T + 168222 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 316 T + 224922 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 235112 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 344 T + 358734 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 416 T + 419608 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 184 T + 574078 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 64 T + 704678 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1052 T + 809710 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 668 T + 1087834 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 68 T + 840530 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 468 T + 1318894 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 444 T + 620102 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.779785897911167683040466791550, −8.479042014239763599117093972148, −8.321849542673526551939410982071, −7.917729052537548775644434941620, −7.44588481357580159650206021559, −7.21540277290606944356800222118, −6.57808346500261924244310345301, −6.55211077311833451801489099963, −5.55507902239688055635784684635, −5.43815176401465270082964340457, −4.43011357653329757317756362873, −4.21493838202730050009554821129, −3.52356603276467525168971168330, −3.27434620099753897524482864912, −2.51497033910384236245797463782, −2.34574950943230043540063162745, −1.33556397557428726290086073268, −1.26661667115964919608025712018, 0, 0,
1.26661667115964919608025712018, 1.33556397557428726290086073268, 2.34574950943230043540063162745, 2.51497033910384236245797463782, 3.27434620099753897524482864912, 3.52356603276467525168971168330, 4.21493838202730050009554821129, 4.43011357653329757317756362873, 5.43815176401465270082964340457, 5.55507902239688055635784684635, 6.55211077311833451801489099963, 6.57808346500261924244310345301, 7.21540277290606944356800222118, 7.44588481357580159650206021559, 7.917729052537548775644434941620, 8.321849542673526551939410982071, 8.479042014239763599117093972148, 8.779785897911167683040466791550