| L(s) = 1 | + 16·2-s + 56·3-s + 192·4-s + 238·5-s + 896·6-s + 2.04e3·8-s + 367·9-s + 3.80e3·10-s + 5.84e3·11-s + 1.07e4·12-s − 1.31e3·13-s + 1.33e4·15-s + 2.04e4·16-s + 4.76e4·17-s + 5.87e3·18-s + 4.10e4·19-s + 4.56e4·20-s + 9.35e4·22-s − 4.93e4·23-s + 1.14e5·24-s − 1.04e5·25-s − 2.10e4·26-s − 1.20e4·27-s + 1.72e5·29-s + 2.13e5·30-s + 7.02e4·31-s + 1.96e5·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.19·3-s + 3/2·4-s + 0.851·5-s + 1.69·6-s + 1.41·8-s + 0.167·9-s + 1.20·10-s + 1.32·11-s + 1.79·12-s − 0.166·13-s + 1.01·15-s + 5/4·16-s + 2.35·17-s + 0.237·18-s + 1.37·19-s + 1.27·20-s + 1.87·22-s − 0.845·23-s + 1.69·24-s − 1.33·25-s − 0.234·26-s − 0.117·27-s + 1.31·29-s + 1.44·30-s + 0.423·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(18.71670003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.71670003\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 56 T + 923 p T^{2} - 56 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 238 T + 32171 p T^{2} - 238 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5848 T + 47407057 T^{2} - 5848 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1316 T + 34154174 T^{2} + 1316 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 47642 T + 1226620987 T^{2} - 47642 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 41048 T + 2067909993 T^{2} - 41048 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 49316 T + 5703777757 T^{2} + 49316 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 172820 T + 40166510782 T^{2} - 172820 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 70252 T + 27708485837 T^{2} - 70252 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 88166 T - 95268180561 T^{2} + 88166 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1156260 T + 721703173798 T^{2} - 1156260 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 57544 T + 389112928854 T^{2} - 57544 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1412292 T + 1262605282717 T^{2} - 1412292 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2361174 T + 3734674197343 T^{2} + 2361174 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1842512 T + 5249017902265 T^{2} + 1842512 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1278242 T + 4493399672183 T^{2} + 1278242 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2121480 T + 8008964895265 T^{2} + 2121480 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3600160 T + 20950979040046 T^{2} - 3600160 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1634682 T + 16729988837419 T^{2} - 1634682 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9192604 T + 59375476967813 T^{2} + 9192604 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 28280 T + 42919362781510 T^{2} - 28280 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8936550 T + 104148685455019 T^{2} + 8936550 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5874204 T + 167733185963446 T^{2} + 5874204 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.68128995820841760049113027627, −12.47081475771959736760449757338, −11.85541542835129062831177530264, −11.59752989319826970808790626767, −10.61499856601316767402627559887, −10.04651109140108529194392123578, −9.372830083780980812941048813524, −9.316843931078176135365164878668, −7.984567752194569375783974300840, −7.910585191162579830923012563750, −7.14949335663912403044164410286, −6.18946242750792278272848347253, −5.86967758712021433427887910180, −5.33405921288633233049068553260, −4.27153298183021187708993559745, −3.80263363262133523616994091219, −2.90848645658588612329158493966, −2.74661911354849801807531505003, −1.57303650202418092760625594517, −1.12111117009806743020517413200,
1.12111117009806743020517413200, 1.57303650202418092760625594517, 2.74661911354849801807531505003, 2.90848645658588612329158493966, 3.80263363262133523616994091219, 4.27153298183021187708993559745, 5.33405921288633233049068553260, 5.86967758712021433427887910180, 6.18946242750792278272848347253, 7.14949335663912403044164410286, 7.910585191162579830923012563750, 7.984567752194569375783974300840, 9.316843931078176135365164878668, 9.372830083780980812941048813524, 10.04651109140108529194392123578, 10.61499856601316767402627559887, 11.59752989319826970808790626767, 11.85541542835129062831177530264, 12.47081475771959736760449757338, 12.68128995820841760049113027627
