| L(s) = 1 | − 3-s + 5-s − 2·9-s − 6·13-s − 15-s − 2·17-s − 6·19-s − 6·25-s + 2·27-s − 5·31-s + 2·37-s + 6·39-s + 3·41-s + 11·43-s − 2·45-s − 2·47-s + 2·51-s − 11·53-s + 6·57-s − 4·59-s + 5·61-s − 6·65-s − 7·67-s + 6·71-s + 5·73-s + 6·75-s + 18·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.66·13-s − 0.258·15-s − 0.485·17-s − 1.37·19-s − 6/5·25-s + 0.384·27-s − 0.898·31-s + 0.328·37-s + 0.960·39-s + 0.468·41-s + 1.67·43-s − 0.298·45-s − 0.291·47-s + 0.280·51-s − 1.51·53-s + 0.794·57-s − 0.520·59-s + 0.640·61-s − 0.744·65-s − 0.855·67-s + 0.712·71-s + 0.585·73-s + 0.692·75-s + 2.02·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 65 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 113 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 133 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 143 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 26 T + 334 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.310230596592226970569694608888, −8.080519793118448260531075931618, −7.60429034101457519171813928042, −7.47951650687549184080147174118, −6.77275661857174000064795688344, −6.59303093692606308591028115367, −5.99793630692987909730304788233, −5.99304240476876300710335641998, −5.27993423962951992515512820488, −5.21708031830030720063256730280, −4.46988949777665699571497919437, −4.42010476337939871781300369976, −3.74340661881217682337302857514, −3.30186685727323101955848324798, −2.52195557534176132691921717628, −2.32159944497486696868374542878, −1.98685144373014751336711921079, −1.11271014048077772667575687308, 0, 0,
1.11271014048077772667575687308, 1.98685144373014751336711921079, 2.32159944497486696868374542878, 2.52195557534176132691921717628, 3.30186685727323101955848324798, 3.74340661881217682337302857514, 4.42010476337939871781300369976, 4.46988949777665699571497919437, 5.21708031830030720063256730280, 5.27993423962951992515512820488, 5.99304240476876300710335641998, 5.99793630692987909730304788233, 6.59303093692606308591028115367, 6.77275661857174000064795688344, 7.47951650687549184080147174118, 7.60429034101457519171813928042, 8.080519793118448260531075931618, 8.310230596592226970569694608888
