| L(s) = 1 | − 5.27·2-s − 3·3-s + 19.8·4-s − 10.5·5-s + 15.8·6-s − 62.3·8-s + 9·9-s + 55.6·10-s + 34.7·11-s − 59.4·12-s + 37.2·13-s + 31.6·15-s + 170.·16-s + 10.5·17-s − 47.4·18-s + 58.5·19-s − 209.·20-s − 183.·22-s − 125.·23-s + 187.·24-s − 13.7·25-s − 196.·26-s − 27·27-s − 35.4·29-s − 166.·30-s − 291.·31-s − 399.·32-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.943·5-s + 1.07·6-s − 2.75·8-s + 0.333·9-s + 1.75·10-s + 0.952·11-s − 1.43·12-s + 0.795·13-s + 0.544·15-s + 2.66·16-s + 0.150·17-s − 0.621·18-s + 0.707·19-s − 2.33·20-s − 1.77·22-s − 1.13·23-s + 1.59·24-s − 0.109·25-s − 1.48·26-s − 0.192·27-s − 0.226·29-s − 1.01·30-s − 1.69·31-s − 2.20·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.27T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.54165661381019077906934281816, −11.09866397759137888896345185874, −9.919041903214405352448015738992, −8.976583764827087722049569487567, −7.942311569089906497242438401278, −7.09375136027786462666250626686, −5.96384056423823099496780510937, −3.70498618959406628522758601989, −1.48348065498068998530169003535, 0,
1.48348065498068998530169003535, 3.70498618959406628522758601989, 5.96384056423823099496780510937, 7.09375136027786462666250626686, 7.942311569089906497242438401278, 8.976583764827087722049569487567, 9.919041903214405352448015738992, 11.09866397759137888896345185874, 11.54165661381019077906934281816
