| L(s) = 1 | − 3-s + 2·5-s − 7-s − 9-s − 4·11-s − 3·13-s − 2·15-s + 17-s − 12·19-s + 21-s − 2·23-s + 3·25-s + 4·29-s + 2·31-s + 4·33-s − 2·35-s − 5·37-s + 3·39-s − 2·45-s + 3·47-s − 9·49-s − 51-s − 5·53-s − 8·55-s + 12·57-s − 5·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 1/3·9-s − 1.20·11-s − 0.832·13-s − 0.516·15-s + 0.242·17-s − 2.75·19-s + 0.218·21-s − 0.417·23-s + 3/5·25-s + 0.742·29-s + 0.359·31-s + 0.696·33-s − 0.338·35-s − 0.821·37-s + 0.480·39-s − 0.298·45-s + 0.437·47-s − 9/7·49-s − 0.140·51-s − 0.686·53-s − 1.07·55-s + 1.58·57-s − 0.650·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 172 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 161 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 184 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.901544880561462316381874466877, −8.793849388000070594543696754177, −8.159414376033163239288205084303, −8.093801949440085159982244378357, −7.38307302199769513458956062097, −7.02656772355351363048196477379, −6.48470323780968505175660361376, −6.29133643802436696104142495865, −5.70038962179969019640991802601, −5.69588740487162721821107475148, −4.85738872382545566253811259373, −4.72781233605554511483273136684, −4.24864254885099928679391452767, −3.54242697742674730233247379829, −2.79652366704714863587821808439, −2.62532269527398296788564353162, −2.01861015632329204615327054359, −1.44146126821444297940418773491, 0, 0,
1.44146126821444297940418773491, 2.01861015632329204615327054359, 2.62532269527398296788564353162, 2.79652366704714863587821808439, 3.54242697742674730233247379829, 4.24864254885099928679391452767, 4.72781233605554511483273136684, 4.85738872382545566253811259373, 5.69588740487162721821107475148, 5.70038962179969019640991802601, 6.29133643802436696104142495865, 6.48470323780968505175660361376, 7.02656772355351363048196477379, 7.38307302199769513458956062097, 8.093801949440085159982244378357, 8.159414376033163239288205084303, 8.793849388000070594543696754177, 8.901544880561462316381874466877
