| L(s) = 1 | − 2-s − 2·3-s + 2·6-s − 5·7-s + 8-s + 3·9-s − 4·11-s + 7·13-s + 5·14-s − 16-s − 4·17-s − 3·18-s + 2·19-s + 10·21-s + 4·22-s + 4·23-s − 2·24-s + 5·25-s − 7·26-s − 4·27-s − 6·29-s + 8·33-s + 4·34-s + 11·37-s − 2·38-s − 14·39-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 0.816·6-s − 1.88·7-s + 0.353·8-s + 9-s − 1.20·11-s + 1.94·13-s + 1.33·14-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.458·19-s + 2.18·21-s + 0.852·22-s + 0.834·23-s − 0.408·24-s + 25-s − 1.37·26-s − 0.769·27-s − 1.11·29-s + 1.39·33-s + 0.685·34-s + 1.80·37-s − 0.324·38-s − 2.24·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4068775200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4068775200\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−10.96505493948153775505026564181, −10.63808156399107579626554040805, −10.14656499049722708863401473638, −9.716450555996557231814114486217, −9.345631333390841040573224792489, −8.971174371970919606599974057965, −8.426439441411713952972488830057, −7.967480470006310069205083854215, −7.27417136595214622397978430678, −6.92525076046546996465140001714, −6.40560230462342402064102938847, −6.16760027991570184922808579047, −5.39310197254778900217970881164, −5.33585081303788775726538513519, −4.17572968269952603645565269157, −4.02128339776417857138440775527, −3.06626493886413318983046894820, −2.64311705229149036855107725981, −1.33290728139460089035779235365, −0.48937629211104165898280120423,
0.48937629211104165898280120423, 1.33290728139460089035779235365, 2.64311705229149036855107725981, 3.06626493886413318983046894820, 4.02128339776417857138440775527, 4.17572968269952603645565269157, 5.33585081303788775726538513519, 5.39310197254778900217970881164, 6.16760027991570184922808579047, 6.40560230462342402064102938847, 6.92525076046546996465140001714, 7.27417136595214622397978430678, 7.967480470006310069205083854215, 8.426439441411713952972488830057, 8.971174371970919606599974057965, 9.345631333390841040573224792489, 9.716450555996557231814114486217, 10.14656499049722708863401473638, 10.63808156399107579626554040805, 10.96505493948153775505026564181
