| L(s) = 1 | − 2·3-s + 4-s − 6·5-s − 8·7-s + 4·9-s + 12·11-s − 2·12-s + 12·15-s − 6·17-s + 6·19-s − 6·20-s + 16·21-s + 11·25-s − 4·27-s − 8·28-s − 24·33-s + 48·35-s + 4·36-s − 2·37-s + 6·41-s + 12·44-s − 24·45-s + 12·47-s + 30·49-s + 12·51-s + 12·53-s − 72·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 2.68·5-s − 3.02·7-s + 4/3·9-s + 3.61·11-s − 0.577·12-s + 3.09·15-s − 1.45·17-s + 1.37·19-s − 1.34·20-s + 3.49·21-s + 11/5·25-s − 0.769·27-s − 1.51·28-s − 4.17·33-s + 8.11·35-s + 2/3·36-s − 0.328·37-s + 0.937·41-s + 1.80·44-s − 3.57·45-s + 1.75·47-s + 30/7·49-s + 1.68·51-s + 1.64·53-s − 9.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2908499195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2908499195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 43 T^{2} + 18 T^{3} + 3084 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 14 T^{2} - 288 T^{3} + 6459 T^{4} - 288 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 142 T^{2} - 1128 T^{3} + 8187 T^{4} - 1128 p T^{5} + 142 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T - 104 T^{2} - 52 T^{3} + 6907 T^{4} - 52 p T^{5} - 104 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T - 22 T^{2} + 288 T^{3} + 12291 T^{4} + 288 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 284 T^{2} + 3168 T^{3} + 33267 T^{4} + 3168 p T^{5} + 284 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 396 T^{3} + 123 T^{4} - 396 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T + 116 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23766587534515271438891961803, −10.50272125885903166084515845387, −10.19986489249041837987230741306, −9.986894660542854734188828077346, −9.853175036921629171389176806789, −9.202167414591515130382824437452, −9.110551508020049280683057617745, −8.811045922484158659445534165781, −8.675113731009073212394740525707, −7.967186685140303605829202523961, −7.36174003860689635232227643703, −7.19029689985579873610718444027, −7.12647051850906816776564857636, −6.78100534759549792262131656320, −6.47761877574681070624724372865, −6.20445608735266425771530006515, −5.88587712832323210442287172801, −5.41074817060273860177902105043, −4.35454702863015530219870428384, −4.35134283526187355372783058123, −3.84129585771287123298936841095, −3.58944060761563235987323413755, −3.55463486096395202303844043097, −2.51376875735871320420041919262, −0.858817157541367596862807090764,
0.858817157541367596862807090764, 2.51376875735871320420041919262, 3.55463486096395202303844043097, 3.58944060761563235987323413755, 3.84129585771287123298936841095, 4.35134283526187355372783058123, 4.35454702863015530219870428384, 5.41074817060273860177902105043, 5.88587712832323210442287172801, 6.20445608735266425771530006515, 6.47761877574681070624724372865, 6.78100534759549792262131656320, 7.12647051850906816776564857636, 7.19029689985579873610718444027, 7.36174003860689635232227643703, 7.967186685140303605829202523961, 8.675113731009073212394740525707, 8.811045922484158659445534165781, 9.110551508020049280683057617745, 9.202167414591515130382824437452, 9.853175036921629171389176806789, 9.986894660542854734188828077346, 10.19986489249041837987230741306, 10.50272125885903166084515845387, 11.23766587534515271438891961803
