| L(s) = 1 | + 4.12·2-s − 3·3-s + 9·4-s − 3.05·5-s − 12.3·6-s − 6.68·7-s + 4.12·8-s + 9·9-s − 12.5·10-s + 32.2·11-s − 27·12-s − 27.5·14-s + 9.15·15-s − 55·16-s − 28.8·17-s + 37.1·18-s − 101.·19-s − 27.4·20-s + 20.0·21-s + 132.·22-s − 118.·23-s − 12.3·24-s − 115.·25-s − 27·27-s − 60.1·28-s + 160.·29-s + 37.7·30-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.12·4-s − 0.272·5-s − 0.841·6-s − 0.360·7-s + 0.182·8-s + 0.333·9-s − 0.397·10-s + 0.883·11-s − 0.649·12-s − 0.526·14-s + 0.157·15-s − 0.859·16-s − 0.411·17-s + 0.485·18-s − 1.22·19-s − 0.306·20-s + 0.208·21-s + 1.28·22-s − 1.07·23-s − 0.105·24-s − 0.925·25-s − 0.192·27-s − 0.405·28-s + 1.02·29-s + 0.229·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.12T + 8T^{2} \) |
| 5 | \( 1 + 3.05T + 125T^{2} \) |
| 7 | \( 1 + 6.68T + 343T^{2} \) |
| 11 | \( 1 - 32.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 38.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 56.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 517.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 656.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 57.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 687.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.75e3T + 9.12e5T^{2} \) |
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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
−10.32615474337927305738333369340, −9.257896037209184174127835388791, −8.109467081423732818915629015410, −6.58592266294271919501154111480, −6.39703673118474459700452791099, −5.19297283451489861832987095013, −4.24676278489650768344372372894, −3.54595320269453940479262184079, −2.02672685350344045211732700666, 0,
2.02672685350344045211732700666, 3.54595320269453940479262184079, 4.24676278489650768344372372894, 5.19297283451489861832987095013, 6.39703673118474459700452791099, 6.58592266294271919501154111480, 8.109467081423732818915629015410, 9.257896037209184174127835388791, 10.32615474337927305738333369340
