| L(s) = 1 | + 2·2-s − 3·3-s − 2·5-s − 6·6-s − 42·7-s − 8·8-s − 4·10-s − 80·11-s − 17·13-s − 84·14-s + 6·15-s − 16·16-s − 36·17-s − 152·19-s + 126·21-s − 160·22-s − 74·23-s + 24·24-s + 125·25-s − 34·26-s + 27·27-s − 100·29-s + 12·30-s + 206·31-s + 240·33-s − 72·34-s + 84·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.178·5-s − 0.408·6-s − 2.26·7-s − 0.353·8-s − 0.126·10-s − 2.19·11-s − 0.362·13-s − 1.60·14-s + 0.103·15-s − 1/4·16-s − 0.513·17-s − 1.83·19-s + 1.30·21-s − 1.55·22-s − 0.670·23-s + 0.204·24-s + 25-s − 0.256·26-s + 0.192·27-s − 0.640·29-s + 0.0730·30-s + 1.19·31-s + 1.26·33-s − 0.363·34-s + 0.405·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 p T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 40 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 17 T - 1908 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 36 T - 3617 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 74 T - 6691 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 100 T - 14389 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 103 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 187 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 128 T - 52537 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 121 T - 64866 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 410 T + 64277 T^{2} + 410 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 230 T - 95977 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 744 T + 348157 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 277 T - 150252 T^{2} - 277 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 231 T - 247402 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 578 T - 23827 T^{2} + 578 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 609 T - 18136 T^{2} + 609 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1259 T + 1092042 T^{2} + 1259 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 696 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 612 T - 330425 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1550 T + 1489827 T^{2} - 1550 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.94741871954418398059413206255, −12.73916650378305619495854572218, −11.94697761180547833740940030668, −11.38123570463625269226395067414, −10.57856798386827375614239409340, −10.38358204948418153651095580219, −9.765108478447073019015467603586, −9.304550973083164130757031955570, −8.332502146744416987728103523825, −8.028796222208092094559783516480, −6.90912872215452073318239733632, −6.60166823859674941996319659563, −5.98090909886034794711634773996, −5.44064854089498148039607764008, −4.61257630574850714318613260702, −4.00004182325913001412303918894, −2.79763033543665769729711732584, −2.69008936743913888765634550373, 0, 0,
2.69008936743913888765634550373, 2.79763033543665769729711732584, 4.00004182325913001412303918894, 4.61257630574850714318613260702, 5.44064854089498148039607764008, 5.98090909886034794711634773996, 6.60166823859674941996319659563, 6.90912872215452073318239733632, 8.028796222208092094559783516480, 8.332502146744416987728103523825, 9.304550973083164130757031955570, 9.765108478447073019015467603586, 10.38358204948418153651095580219, 10.57856798386827375614239409340, 11.38123570463625269226395067414, 11.94697761180547833740940030668, 12.73916650378305619495854572218, 12.94741871954418398059413206255
