| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 3·9-s − 3·11-s − 2·13-s − 2·15-s − 3·17-s − 9·19-s + 4·21-s + 3·23-s − 5·25-s + 4·27-s − 7·29-s − 4·31-s − 6·33-s − 2·35-s − 11·37-s − 4·39-s + 8·41-s − 3·43-s − 3·45-s − 2·47-s + 3·49-s − 6·51-s + 3·55-s − 18·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 9-s − 0.904·11-s − 0.554·13-s − 0.516·15-s − 0.727·17-s − 2.06·19-s + 0.872·21-s + 0.625·23-s − 25-s + 0.769·27-s − 1.29·29-s − 0.718·31-s − 1.04·33-s − 0.338·35-s − 1.80·37-s − 0.640·39-s + 1.24·41-s − 0.457·43-s − 0.447·45-s − 0.291·47-s + 3/7·49-s − 0.840·51-s + 0.404·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 54 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 22 T + 222 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 148 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 22 T + 238 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.995841410794605657385954995389, −7.916228474902131866764908693513, −7.50644068485940023481948578737, −7.38625449350990735794626943133, −6.82132767612748523601638895262, −6.52048228417254395148596106053, −5.78795207891730008284500480070, −5.78627272912747127760302293925, −5.01263482172132695116348828358, −4.74958373853040803429152519446, −4.25706881619517796296866321487, −4.19747626504083924060128726148, −3.43079965559702342031619757136, −3.26636903849464476919044965066, −2.54849015474497376084636506837, −2.23286820845543836649742748795, −1.82134799430498044776512891717, −1.44174314432941352836006377215, 0, 0,
1.44174314432941352836006377215, 1.82134799430498044776512891717, 2.23286820845543836649742748795, 2.54849015474497376084636506837, 3.26636903849464476919044965066, 3.43079965559702342031619757136, 4.19747626504083924060128726148, 4.25706881619517796296866321487, 4.74958373853040803429152519446, 5.01263482172132695116348828358, 5.78627272912747127760302293925, 5.78795207891730008284500480070, 6.52048228417254395148596106053, 6.82132767612748523601638895262, 7.38625449350990735794626943133, 7.50644068485940023481948578737, 7.916228474902131866764908693513, 7.995841410794605657385954995389
