| L(s) = 1 | + 4·3-s + 4·7-s + 8·9-s − 4·11-s + 4·13-s + 4·17-s − 2·19-s + 16·21-s + 12·23-s + 12·27-s + 4·29-s − 8·31-s − 16·33-s + 12·37-s + 16·39-s − 4·41-s − 4·43-s + 4·47-s + 6·49-s + 16·51-s − 4·53-s − 8·57-s + 16·59-s − 16·61-s + 32·63-s + 4·67-s + 48·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.51·7-s + 8/3·9-s − 1.20·11-s + 1.10·13-s + 0.970·17-s − 0.458·19-s + 3.49·21-s + 2.50·23-s + 2.30·27-s + 0.742·29-s − 1.43·31-s − 2.78·33-s + 1.97·37-s + 2.56·39-s − 0.624·41-s − 0.609·43-s + 0.583·47-s + 6/7·49-s + 2.24·51-s − 0.549·53-s − 1.05·57-s + 2.08·59-s − 2.04·61-s + 4.03·63-s + 0.488·67-s + 5.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.503530624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.503530624\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 212 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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
−9.105350401243179652686633724661, −8.991426195530261133363842420588, −8.518153926659759368142620055655, −8.286866276828365740360376758947, −7.895962890627497027233527050110, −7.82274375813984079578170040733, −7.12570631103319291875353444803, −7.07776387859571917303056921611, −6.22801289309547539997526798333, −5.80048231792358514132792270165, −5.09402850330044237063682818958, −4.95184551981428610559216145997, −4.50477855732969339547605811814, −3.80120206116264116607526388645, −3.39063017175449050963740697219, −3.10581653026542240712445624896, −2.38876468739221003337205171349, −2.33385823526127338015047033583, −1.38357515986915209515578547703, −1.06366429836805963243822559245,
1.06366429836805963243822559245, 1.38357515986915209515578547703, 2.33385823526127338015047033583, 2.38876468739221003337205171349, 3.10581653026542240712445624896, 3.39063017175449050963740697219, 3.80120206116264116607526388645, 4.50477855732969339547605811814, 4.95184551981428610559216145997, 5.09402850330044237063682818958, 5.80048231792358514132792270165, 6.22801289309547539997526798333, 7.07776387859571917303056921611, 7.12570631103319291875353444803, 7.82274375813984079578170040733, 7.895962890627497027233527050110, 8.286866276828365740360376758947, 8.518153926659759368142620055655, 8.991426195530261133363842420588, 9.105350401243179652686633724661
