| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s + 5-s − 24·6-s − 14·7-s − 32·8-s + 27·9-s − 4·10-s − 16·11-s + 72·12-s − 26·13-s + 56·14-s + 6·15-s + 80·16-s + 18·17-s − 108·18-s + 63·19-s + 12·20-s − 84·21-s + 64·22-s − 95·23-s − 192·24-s − 217·25-s + 104·26-s + 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.0894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.126·10-s − 0.438·11-s + 1.73·12-s − 0.554·13-s + 1.06·14-s + 0.103·15-s + 5/4·16-s + 0.256·17-s − 1.41·18-s + 0.760·19-s + 0.134·20-s − 0.872·21-s + 0.620·22-s − 0.861·23-s − 1.63·24-s − 1.73·25-s + 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 218 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 16 T + 2210 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 18 T + 3586 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 63 T + 5390 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 95 T + 9530 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 419 T + 91088 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 73 T + 60882 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 90 T + 35090 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 186 T + 38002 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 67 T + 136626 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 421 T + 251924 T^{2} + 421 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 819 T + 394204 T^{2} + 819 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 698 T + 531398 T^{2} + 698 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 68 T + 157902 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 638 T + 700062 T^{2} - 638 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 538 T + 766382 T^{2} + 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 541 T + 191724 T^{2} - 541 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 815 T + 1080894 T^{2} + 815 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 291 T + 1157488 T^{2} - 291 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 945 T + 356062 T^{2} + 945 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 23 T + 1634268 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.790123134128536799088259468065, −9.605284750774727919276580183024, −9.307657521557175911444393825178, −9.231213262047884386489035705174, −8.206422999833152166964222667455, −7.967042510896100582807416408608, −7.70787908719662167531816966641, −7.37289794031841749582743307472, −6.66814066786533764008775580892, −6.31235716596756363088837624645, −5.53060053218149427209970626958, −5.30256366122942019045428068996, −3.95680359232056079770773355994, −3.90701783051876407669587130152, −2.93969727675367858733959102684, −2.71370216544147130421061147040, −1.74617647330597940306195420218, −1.57712576309046620530899965683, 0, 0,
1.57712576309046620530899965683, 1.74617647330597940306195420218, 2.71370216544147130421061147040, 2.93969727675367858733959102684, 3.90701783051876407669587130152, 3.95680359232056079770773355994, 5.30256366122942019045428068996, 5.53060053218149427209970626958, 6.31235716596756363088837624645, 6.66814066786533764008775580892, 7.37289794031841749582743307472, 7.70787908719662167531816966641, 7.967042510896100582807416408608, 8.206422999833152166964222667455, 9.231213262047884386489035705174, 9.307657521557175911444393825178, 9.605284750774727919276580183024, 9.790123134128536799088259468065
