| L(s) = 1 | − 2·5-s − 4·7-s − 4·11-s − 2·13-s − 4·17-s + 4·19-s + 8·23-s + 3·25-s − 8·29-s + 4·31-s + 8·35-s − 12·41-s − 8·43-s − 4·47-s − 2·49-s − 12·53-s + 8·55-s + 4·59-s + 8·61-s + 4·65-s + 20·67-s − 12·71-s + 16·77-s + 4·83-s + 8·85-s − 12·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.970·17-s + 0.917·19-s + 1.66·23-s + 3/5·25-s − 1.48·29-s + 0.718·31-s + 1.35·35-s − 1.87·41-s − 1.21·43-s − 0.583·47-s − 2/7·49-s − 1.64·53-s + 1.07·55-s + 0.520·59-s + 1.02·61-s + 0.496·65-s + 2.44·67-s − 1.42·71-s + 1.82·77-s + 0.439·83-s + 0.867·85-s − 1.27·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8266696142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8266696142\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77205440815348014317077570762, −7.70727237184113514608136864128, −6.94662088522018848188914207050, −6.85125252822541008203862417707, −6.63310963620892370828605445276, −6.53255652006910285148653264852, −5.56803387310248850463965791361, −5.50737844115263622899769285868, −5.11228072166909072334801041711, −4.89636875718957182353661540301, −4.32275942810037946211135355664, −4.04653808663558234924783036951, −3.42448266190533901712907782464, −3.24208821229344341166939280816, −2.82630205624995873760665583718, −2.79735866783702788274398438805, −1.83061995344127152210578849019, −1.66697266065152779182307771343, −0.57650592475608153199575526950, −0.34377221654586199444786959032,
0.34377221654586199444786959032, 0.57650592475608153199575526950, 1.66697266065152779182307771343, 1.83061995344127152210578849019, 2.79735866783702788274398438805, 2.82630205624995873760665583718, 3.24208821229344341166939280816, 3.42448266190533901712907782464, 4.04653808663558234924783036951, 4.32275942810037946211135355664, 4.89636875718957182353661540301, 5.11228072166909072334801041711, 5.50737844115263622899769285868, 5.56803387310248850463965791361, 6.53255652006910285148653264852, 6.63310963620892370828605445276, 6.85125252822541008203862417707, 6.94662088522018848188914207050, 7.70727237184113514608136864128, 7.77205440815348014317077570762
