| L(s) = 1 | − 2·2-s + 3·3-s − 5·5-s − 6·6-s − 28·7-s + 8·8-s + 10·10-s + 20·11-s + 32·13-s + 56·14-s − 15·15-s − 16·16-s − 89·17-s + 133·19-s − 84·21-s − 40·22-s − 56·23-s + 24·24-s − 64·26-s − 27·27-s − 210·29-s + 30·30-s + 326·31-s + 60·33-s + 178·34-s + 140·35-s + 136·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.548·11-s + 0.682·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s − 1.26·17-s + 1.60·19-s − 0.872·21-s − 0.387·22-s − 0.507·23-s + 0.204·24-s − 0.482·26-s − 0.192·27-s − 1.34·29-s + 0.182·30-s + 1.88·31-s + 0.316·33-s + 0.897·34-s + 0.676·35-s + 0.604·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8599124322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8599124322\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 32 T - 1173 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 89 T + 3008 T^{2} + 89 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 56 T - 9031 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 163 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 28 T - 68137 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 156 T - 55171 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 135 T - 85598 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 105 T - 137852 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T - 192383 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 758 T + 347583 T^{2} + 758 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 926 T + 556713 T^{2} + 926 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 357847 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T - 387861 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 696 T - 8623 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 319 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 558 T - 393605 T^{2} + 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1710 T + 2011427 T^{2} - 1710 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.37501428813657662853738160463, −9.925514583163052141540187236863, −9.570312351422011289172409444292, −9.171634650945778646688750288146, −9.043559299117553909502396309010, −8.315703948623565727324651469404, −8.066840835756437521557252507510, −7.44077962651006121869195564393, −7.13318509807914173226328716614, −6.51540741265862209045785246405, −6.07923112659275490707589214181, −5.79490908154155265058700890033, −4.66819519867096978671784539485, −4.46653746578495429891648631089, −3.59022096307358660213195744141, −3.34529385899911707086914735230, −2.78042066651959841422499352680, −1.94065337902417157705143925309, −1.11930993324414583196703529299, −0.34037671038791780635367621771,
0.34037671038791780635367621771, 1.11930993324414583196703529299, 1.94065337902417157705143925309, 2.78042066651959841422499352680, 3.34529385899911707086914735230, 3.59022096307358660213195744141, 4.46653746578495429891648631089, 4.66819519867096978671784539485, 5.79490908154155265058700890033, 6.07923112659275490707589214181, 6.51540741265862209045785246405, 7.13318509807914173226328716614, 7.44077962651006121869195564393, 8.066840835756437521557252507510, 8.315703948623565727324651469404, 9.043559299117553909502396309010, 9.171634650945778646688750288146, 9.570312351422011289172409444292, 9.925514583163052141540187236863, 10.37501428813657662853738160463
