| L(s) = 1 | + 4·2-s + 12·3-s + 12·4-s + 10·5-s + 48·6-s + 4·7-s + 32·8-s + 64·9-s + 40·10-s − 24·11-s + 144·12-s − 26·13-s + 16·14-s + 120·15-s + 80·16-s + 28·17-s + 256·18-s − 32·19-s + 120·20-s + 48·21-s − 96·22-s − 164·23-s + 384·24-s + 75·25-s − 104·26-s + 132·27-s + 48·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s + 3.26·6-s + 0.215·7-s + 1.41·8-s + 2.37·9-s + 1.26·10-s − 0.657·11-s + 3.46·12-s − 0.554·13-s + 0.305·14-s + 2.06·15-s + 5/4·16-s + 0.399·17-s + 3.35·18-s − 0.386·19-s + 1.34·20-s + 0.498·21-s − 0.930·22-s − 1.48·23-s + 3.26·24-s + 3/5·25-s − 0.784·26-s + 0.940·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.74435683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.74435683\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 4 p T + 80 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 530 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 24 T + 556 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 28 T + 8062 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 32 T + 7724 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 164 T + 20168 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 136 T + 17402 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 44708 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 88 T + 101282 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 260 T + 152782 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 20 T - 62896 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 228 T + 184642 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 668 T + 404470 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 720 T + 403468 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 136 T + 427226 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 612 T + 695122 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 360 T + 724212 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 680 T + 882074 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1592 T + 1618694 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 300 T + 1066074 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 268 T + 73654 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2636 T + 3462470 T^{2} - 2636 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−13.04068058309326228818187150300, −13.03206085831928070790444218125, −12.42631570168070349096377657994, −11.77868230984975022201710982920, −11.10732222590828365977437236246, −10.38366226948182411589528183805, −9.832572920683859842142064468851, −9.605403853635278430610422179511, −8.594902449748058770494150158051, −8.384141741017600254852383964225, −7.65465702294521922704552872397, −7.36055129933049436659946151651, −6.18908808383133486737851962252, −6.01715350182053170916069567146, −4.82827694822232853422236737392, −4.53667358891281382563386816989, −3.34170506447940766013443343766, −3.09795074208555517402840897222, −2.25975689941869776580707099829, −1.83736881037458674341327054913,
1.83736881037458674341327054913, 2.25975689941869776580707099829, 3.09795074208555517402840897222, 3.34170506447940766013443343766, 4.53667358891281382563386816989, 4.82827694822232853422236737392, 6.01715350182053170916069567146, 6.18908808383133486737851962252, 7.36055129933049436659946151651, 7.65465702294521922704552872397, 8.384141741017600254852383964225, 8.594902449748058770494150158051, 9.605403853635278430610422179511, 9.832572920683859842142064468851, 10.38366226948182411589528183805, 11.10732222590828365977437236246, 11.77868230984975022201710982920, 12.42631570168070349096377657994, 13.03206085831928070790444218125, 13.04068058309326228818187150300
