| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s − 9·11-s + 7·13-s + 4·14-s + 5·16-s − 2·17-s + 5·19-s + 18·22-s + 8·23-s − 10·25-s − 14·26-s − 6·28-s − 7·29-s + 6·31-s − 6·32-s + 4·34-s + 2·37-s − 10·38-s + 8·41-s − 3·43-s − 27·44-s − 16·46-s − 5·47-s + 2·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s − 2.71·11-s + 1.94·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.14·19-s + 3.83·22-s + 1.66·23-s − 2·25-s − 2.74·26-s − 1.13·28-s − 1.29·29-s + 1.07·31-s − 1.06·32-s + 0.685·34-s + 0.328·37-s − 1.62·38-s + 1.24·41-s − 0.457·43-s − 4.07·44-s − 2.35·46-s − 0.729·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 223 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 9 T + 39 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 97 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 139 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 77 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 206 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.247690574750381962274541427881, −7.954662254747251731220146388726, −7.56011509792443073473737329667, −7.50529695556875100698413443252, −6.78821663460158149760108782415, −6.65851719183571248172727110190, −5.98321220193867858619707452242, −5.81994614467151612657437828104, −5.38135476210235001401364753755, −5.21142770384028683772173932314, −4.32702628870538891350220321579, −4.04180875004999774673785202845, −3.21767754619670585181246525905, −3.14643711898661499222983192025, −2.64302441326994689813646973982, −2.27227384306548623543161731604, −1.41361550285983395905450547936, −1.15636574529746908648014457415, 0, 0,
1.15636574529746908648014457415, 1.41361550285983395905450547936, 2.27227384306548623543161731604, 2.64302441326994689813646973982, 3.14643711898661499222983192025, 3.21767754619670585181246525905, 4.04180875004999774673785202845, 4.32702628870538891350220321579, 5.21142770384028683772173932314, 5.38135476210235001401364753755, 5.81994614467151612657437828104, 5.98321220193867858619707452242, 6.65851719183571248172727110190, 6.78821663460158149760108782415, 7.50529695556875100698413443252, 7.56011509792443073473737329667, 7.954662254747251731220146388726, 8.247690574750381962274541427881
