| L(s) = 1 | − 2·2-s − 4·3-s + 3·4-s − 2·5-s + 8·6-s − 4·8-s + 8·9-s + 4·10-s + 4·11-s − 12·12-s + 4·13-s + 8·15-s + 5·16-s − 8·17-s − 16·18-s − 4·19-s − 6·20-s − 8·22-s − 8·23-s + 16·24-s + 3·25-s − 8·26-s − 12·27-s − 4·29-s − 16·30-s − 6·32-s − 16·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/2·4-s − 0.894·5-s + 3.26·6-s − 1.41·8-s + 8/3·9-s + 1.26·10-s + 1.20·11-s − 3.46·12-s + 1.10·13-s + 2.06·15-s + 5/4·16-s − 1.94·17-s − 3.77·18-s − 0.917·19-s − 1.34·20-s − 1.70·22-s − 1.66·23-s + 3.26·24-s + 3/5·25-s − 1.56·26-s − 2.30·27-s − 0.742·29-s − 2.92·30-s − 1.06·32-s − 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−10.73213762512098074802689265877, −10.58610337582269844470310669795, −9.955701569759992037364841241935, −9.515257255811814933643830066743, −8.762847715714262290796433712972, −8.710215133886423869714400237578, −8.027505829786312290029656815937, −7.62368940817791412410796986024, −6.68130388111122230056375947544, −6.67706129260928183873608072134, −6.24121911688125710111233764299, −6.02993191575522903379874152841, −4.99068908344961615257254006412, −4.72405063829918254237162612210, −3.82915150342024593183245811160, −3.53228460490233816266070308549, −1.95349450908375665927514463418, −1.48713854200108292571192289916, 0, 0,
1.48713854200108292571192289916, 1.95349450908375665927514463418, 3.53228460490233816266070308549, 3.82915150342024593183245811160, 4.72405063829918254237162612210, 4.99068908344961615257254006412, 6.02993191575522903379874152841, 6.24121911688125710111233764299, 6.67706129260928183873608072134, 6.68130388111122230056375947544, 7.62368940817791412410796986024, 8.027505829786312290029656815937, 8.710215133886423869714400237578, 8.762847715714262290796433712972, 9.515257255811814933643830066743, 9.955701569759992037364841241935, 10.58610337582269844470310669795, 10.73213762512098074802689265877
