| L(s) = 1 | + 2·3-s − 10·5-s + 16·7-s − 45·9-s + 90·11-s − 50·13-s − 20·15-s − 44·17-s − 108·19-s + 32·21-s + 28·23-s + 41·25-s − 134·27-s + 58·29-s − 66·31-s + 180·33-s − 160·35-s + 40·37-s − 100·39-s + 304·41-s + 130·43-s + 450·45-s + 514·47-s − 110·49-s − 88·51-s − 958·53-s − 900·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.894·5-s + 0.863·7-s − 5/3·9-s + 2.46·11-s − 1.06·13-s − 0.344·15-s − 0.627·17-s − 1.30·19-s + 0.332·21-s + 0.253·23-s + 0.327·25-s − 0.955·27-s + 0.371·29-s − 0.382·31-s + 0.949·33-s − 0.772·35-s + 0.177·37-s − 0.410·39-s + 1.15·41-s + 0.461·43-s + 1.49·45-s + 1.59·47-s − 0.320·49-s − 0.241·51-s − 2.48·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.637320661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637320661\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 59 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 366 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 90 T + 4633 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4635 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 44 T + 8774 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 108 T + 16538 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 28 T - 11974 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T - 10615 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 40 T + 101610 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 304 T + 122546 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 130 T + 146385 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 514 T + 214889 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 958 T + 510971 T^{2} + 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 180 T + 43858 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1028 T + 717294 T^{2} - 1028 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 912 T + 807062 T^{2} - 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 796 T + 867290 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 856 T + 775362 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 318 T + 229433 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1828 T + 1970306 T^{2} - 1828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 944 T + 1601618 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 368 T + 1799202 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−10.87902033551551896557764093067, −10.79188765264643670074588831676, −9.716694473680618885746679341343, −9.361705426817524089501810848018, −8.980787959131185152213587447998, −8.652564576283513742984742549302, −8.058128312120464025977927412444, −7.982433306738535312818840939114, −7.01960784468949715470971004602, −6.89938613727706560447788576810, −6.13066591494045318039692829343, −5.84144626077110379918601811544, −4.91683029115285904396959059690, −4.56967589476089197225407211820, −3.88831739288704220426840972309, −3.66930167293221575639287909652, −2.65112274932986820235599208097, −2.26183433408648421913731492040, −1.33411931446612388888954597732, −0.41383626120473325099055420520,
0.41383626120473325099055420520, 1.33411931446612388888954597732, 2.26183433408648421913731492040, 2.65112274932986820235599208097, 3.66930167293221575639287909652, 3.88831739288704220426840972309, 4.56967589476089197225407211820, 4.91683029115285904396959059690, 5.84144626077110379918601811544, 6.13066591494045318039692829343, 6.89938613727706560447788576810, 7.01960784468949715470971004602, 7.982433306738535312818840939114, 8.058128312120464025977927412444, 8.652564576283513742984742549302, 8.980787959131185152213587447998, 9.361705426817524089501810848018, 9.716694473680618885746679341343, 10.79188765264643670074588831676, 10.87902033551551896557764093067
