| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s − 4·11-s + 8·13-s + 5·16-s + 4·17-s + 4·19-s − 6·20-s − 8·22-s − 4·23-s + 3·25-s + 16·26-s − 8·29-s + 12·31-s + 6·32-s + 8·34-s − 4·37-s + 8·38-s − 8·40-s + 4·41-s − 12·44-s − 8·46-s + 8·47-s + 6·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s − 1.20·11-s + 2.21·13-s + 5/4·16-s + 0.970·17-s + 0.917·19-s − 1.34·20-s − 1.70·22-s − 0.834·23-s + 3/5·25-s + 3.13·26-s − 1.48·29-s + 2.15·31-s + 1.06·32-s + 1.37·34-s − 0.657·37-s + 1.29·38-s − 1.26·40-s + 0.624·41-s − 1.80·44-s − 1.17·46-s + 1.16·47-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.574850520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.574850520\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.303987024085661343858245949819, −8.040264317826222713448324677190, −7.70046186847754064570118579352, −7.60964034759462607285779815898, −6.87595568515737267107206846034, −6.81059713820347139985614417602, −6.08431808317092472635219522169, −5.88453549322163109343573892226, −5.68931913686028594189570537127, −5.12704258308494886166040579866, −4.90931216342004858491689244602, −4.25234001293073340613374062980, −4.04385607359466721433746586888, −3.56138586346781589082217564499, −3.24010507312999346875706925640, −3.09723140342421848408482738148, −2.13105282539703761973567226224, −2.08186429731661302339495601772, −0.907680181253244279289556192720, −0.853874032457850781213992552000,
0.853874032457850781213992552000, 0.907680181253244279289556192720, 2.08186429731661302339495601772, 2.13105282539703761973567226224, 3.09723140342421848408482738148, 3.24010507312999346875706925640, 3.56138586346781589082217564499, 4.04385607359466721433746586888, 4.25234001293073340613374062980, 4.90931216342004858491689244602, 5.12704258308494886166040579866, 5.68931913686028594189570537127, 5.88453549322163109343573892226, 6.08431808317092472635219522169, 6.81059713820347139985614417602, 6.87595568515737267107206846034, 7.60964034759462607285779815898, 7.70046186847754064570118579352, 8.040264317826222713448324677190, 8.303987024085661343858245949819
