| L(s) = 1 | − 3·2-s + 8·4-s − 3·5-s − 7·7-s − 45·8-s + 9·10-s − 15·11-s − 128·13-s + 21·14-s + 135·16-s + 84·17-s + 16·19-s − 24·20-s + 45·22-s − 84·23-s + 125·25-s + 384·26-s − 56·28-s + 594·29-s + 253·31-s − 360·32-s − 252·34-s + 21·35-s + 316·37-s − 48·38-s + 135·40-s − 720·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 4-s − 0.268·5-s − 0.377·7-s − 1.98·8-s + 0.284·10-s − 0.411·11-s − 2.73·13-s + 0.400·14-s + 2.10·16-s + 1.19·17-s + 0.193·19-s − 0.268·20-s + 0.436·22-s − 0.761·23-s + 25-s + 2.89·26-s − 0.377·28-s + 3.80·29-s + 1.46·31-s − 1.98·32-s − 1.27·34-s + 0.101·35-s + 1.40·37-s − 0.204·38-s + 0.533·40-s − 2.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6110947162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6110947162\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 297 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 253 T + 34218 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 363 T - 17108 T^{2} - 363 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 477 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 906 T + 115867 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )^{2} \) |
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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
−15.23103664677289981250143323834, −14.34520857597417977910322597573, −13.85626431182820684584894852303, −12.57850558488342570499673910701, −12.41663850427594300802213263994, −11.83379622712066472602735915478, −11.67947089290166356129346565283, −10.16448849365863446054792451355, −10.12939170901032552824418014413, −9.857748596758354895774400295598, −8.893526795560829037851986070013, −8.147107920806730178073587235006, −7.88830335050283914168109251131, −6.71351177861983897343153583355, −6.65971297350629044580055596238, −5.39935078514453368454433006921, −4.70459420574296370063987558869, −2.93789211719097433935532809527, −2.71962746093523372546122913870, −0.62473778199903321829856653931,
0.62473778199903321829856653931, 2.71962746093523372546122913870, 2.93789211719097433935532809527, 4.70459420574296370063987558869, 5.39935078514453368454433006921, 6.65971297350629044580055596238, 6.71351177861983897343153583355, 7.88830335050283914168109251131, 8.147107920806730178073587235006, 8.893526795560829037851986070013, 9.857748596758354895774400295598, 10.12939170901032552824418014413, 10.16448849365863446054792451355, 11.67947089290166356129346565283, 11.83379622712066472602735915478, 12.41663850427594300802213263994, 12.57850558488342570499673910701, 13.85626431182820684584894852303, 14.34520857597417977910322597573, 15.23103664677289981250143323834
