| L(s) = 1 | + 4·2-s + 3·3-s + 8·4-s + 18·5-s + 12·6-s + 9·9-s + 72·10-s − 50·11-s + 24·12-s − 36·13-s + 54·15-s − 64·16-s + 126·17-s + 36·18-s − 72·19-s + 144·20-s − 200·22-s + 14·23-s + 199·25-s − 144·26-s + 27·27-s + 158·29-s + 216·30-s − 36·31-s − 256·32-s − 150·33-s + 504·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 1.60·5-s + 0.816·6-s + 1/3·9-s + 2.27·10-s − 1.37·11-s + 0.577·12-s − 0.768·13-s + 0.929·15-s − 16-s + 1.79·17-s + 0.471·18-s − 0.869·19-s + 1.60·20-s − 1.93·22-s + 0.126·23-s + 1.59·25-s − 1.08·26-s + 0.192·27-s + 1.01·29-s + 1.31·30-s − 0.208·31-s − 1.41·32-s − 0.791·33-s + 2.54·34-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
| \(L(2)\) |
\(\approx\) |
\(4.555573486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.555573486\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 + 72 T + p^{3} T^{2} \) |
| 23 | \( 1 - 14 T + p^{3} T^{2} \) |
| 29 | \( 1 - 158 T + p^{3} T^{2} \) |
| 31 | \( 1 + 36 T + p^{3} T^{2} \) |
| 37 | \( 1 + 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 270 T + p^{3} T^{2} \) |
| 43 | \( 1 + 324 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 468 T + p^{3} T^{2} \) |
| 61 | \( 1 - 792 T + p^{3} T^{2} \) |
| 67 | \( 1 - 232 T + p^{3} T^{2} \) |
| 71 | \( 1 + 734 T + p^{3} T^{2} \) |
| 73 | \( 1 - 180 T + p^{3} T^{2} \) |
| 79 | \( 1 - 236 T + p^{3} T^{2} \) |
| 83 | \( 1 - 36 T + p^{3} T^{2} \) |
| 89 | \( 1 - 234 T + p^{3} T^{2} \) |
| 97 | \( 1 - 468 T + p^{3} T^{2} \) |
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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
−12.96843986344544735578528046431, −12.12552147704444864962281457413, −10.38643344066472782100784012380, −9.787279203593671244661157272353, −8.368738239451757095261183610432, −6.85556830731515856466521005848, −5.60925462807968588684740756970, −4.97177295071356597367835356982, −3.16256979228540412994003964864, −2.16535761948169951134191111346,
2.16535761948169951134191111346, 3.16256979228540412994003964864, 4.97177295071356597367835356982, 5.60925462807968588684740756970, 6.85556830731515856466521005848, 8.368738239451757095261183610432, 9.787279203593671244661157272353, 10.38643344066472782100784012380, 12.12552147704444864962281457413, 12.96843986344544735578528046431
