| L(s) = 1 | − 3·2-s − 6·3-s + 5·4-s − 6·5-s + 18·6-s − 27·8-s + 27·9-s + 18·10-s − 6·11-s − 30·12-s − 16·13-s + 36·15-s + 69·16-s + 6·17-s − 81·18-s − 64·19-s − 30·20-s + 18·22-s + 6·23-s + 162·24-s − 166·25-s + 48·26-s − 108·27-s − 252·29-s − 108·30-s − 40·31-s − 27·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.15·3-s + 5/8·4-s − 0.536·5-s + 1.22·6-s − 1.19·8-s + 9-s + 0.569·10-s − 0.164·11-s − 0.721·12-s − 0.341·13-s + 0.619·15-s + 1.07·16-s + 0.0856·17-s − 1.06·18-s − 0.772·19-s − 0.335·20-s + 0.174·22-s + 0.0543·23-s + 1.37·24-s − 1.32·25-s + 0.362·26-s − 0.769·27-s − 1.61·29-s − 0.657·30-s − 0.231·31-s − 0.149·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + p^{2} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 6 T + 202 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 9778 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 64 T + 6534 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T - 13890 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 450 T + 175642 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 141790 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 804 T + 380614 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 428 T + 425886 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1072 T + 1063278 T^{2} + 1072 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1944 T + 1957030 T^{2} + 1944 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 366 T + 1156090 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.49369016705596710812253083529, −11.54165661381019077906934281816, −11.19155757729894946452299799359, −11.09866397759137888896345185874, −10.13006655593879205313010021523, −9.919041903214405352448015738992, −9.115301721711439462433924885559, −8.976583764827087722049569487567, −7.942311569089906497242438401278, −7.67882271614363990335635970708, −7.09375136027786462666250626686, −6.23923261447682743996148289684, −5.96384056423823099496780510937, −5.26929482249951518908250795096, −4.36472161102171773630730392156, −3.70498618959406628522758601989, −2.58214401477367678632689952929, −1.48348065498068998530169003535, 0, 0,
1.48348065498068998530169003535, 2.58214401477367678632689952929, 3.70498618959406628522758601989, 4.36472161102171773630730392156, 5.26929482249951518908250795096, 5.96384056423823099496780510937, 6.23923261447682743996148289684, 7.09375136027786462666250626686, 7.67882271614363990335635970708, 7.942311569089906497242438401278, 8.976583764827087722049569487567, 9.115301721711439462433924885559, 9.919041903214405352448015738992, 10.13006655593879205313010021523, 11.09866397759137888896345185874, 11.19155757729894946452299799359, 11.54165661381019077906934281816, 12.49369016705596710812253083529
