| L(s) = 1 | − 2-s − 2·4-s − 6·5-s + 2·7-s + 3·8-s + 6·10-s − 2·14-s + 16-s − 6·17-s + 8·19-s + 12·20-s + 6·23-s + 17·25-s − 4·28-s + 6·29-s − 4·31-s − 2·32-s + 6·34-s − 12·35-s − 14·37-s − 8·38-s − 18·40-s + 4·41-s + 12·43-s − 6·46-s − 8·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 2.68·5-s + 0.755·7-s + 1.06·8-s + 1.89·10-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 2.68·20-s + 1.25·23-s + 17/5·25-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.353·32-s + 1.02·34-s − 2.02·35-s − 2.30·37-s − 1.29·38-s − 2.84·40-s + 0.624·41-s + 1.82·43-s − 0.884·46-s − 1.16·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 239 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_4$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 197 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.825294725944272329630321802987, −8.585757571462810436009879255801, −8.058256400796899096911666786849, −7.82908027332230258949714246092, −7.48288345709863477604451167449, −7.26211643438293660937196934551, −6.82037519054570710613548596964, −6.29749596895807433501213632443, −5.43622727341398506255599862478, −5.15242393735375889007058716769, −4.59791187167355514203412008026, −4.53259829140616762157624530529, −4.00315152302321806152621032405, −3.65715749229852329518262809154, −3.08155469140220351044600582924, −2.70338263040563367682541094956, −1.47362239576384899567656879781, −1.07768031056860399898182511885, 0, 0,
1.07768031056860399898182511885, 1.47362239576384899567656879781, 2.70338263040563367682541094956, 3.08155469140220351044600582924, 3.65715749229852329518262809154, 4.00315152302321806152621032405, 4.53259829140616762157624530529, 4.59791187167355514203412008026, 5.15242393735375889007058716769, 5.43622727341398506255599862478, 6.29749596895807433501213632443, 6.82037519054570710613548596964, 7.26211643438293660937196934551, 7.48288345709863477604451167449, 7.82908027332230258949714246092, 8.058256400796899096911666786849, 8.585757571462810436009879255801, 8.825294725944272329630321802987
