| L(s) = 1 | + 9·3-s + 91·7-s + 81·9-s − 174·11-s − 785·13-s − 1.79e3·17-s − 925·19-s + 819·21-s + 2.34e3·23-s + 729·27-s − 726·29-s − 811·31-s − 1.56e3·33-s − 7.92e3·37-s − 7.06e3·39-s − 360·41-s + 4.95e3·43-s − 9.90e3·47-s − 8.52e3·49-s − 1.61e4·51-s + 8.29e3·53-s − 8.32e3·57-s + 7.01e3·59-s − 5.14e4·61-s + 7.37e3·63-s − 581·67-s + 2.11e4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.701·7-s + 1/3·9-s − 0.433·11-s − 1.28·13-s − 1.50·17-s − 0.587·19-s + 0.405·21-s + 0.924·23-s + 0.192·27-s − 0.160·29-s − 0.151·31-s − 0.250·33-s − 0.951·37-s − 0.743·39-s − 0.0334·41-s + 0.408·43-s − 0.654·47-s − 0.507·49-s − 0.869·51-s + 0.405·53-s − 0.339·57-s + 0.262·59-s − 1.76·61-s + 0.233·63-s − 0.0158·67-s + 0.533·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 174 T + p^{5} T^{2} \) |
| 13 | \( 1 + 785 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1794 T + p^{5} T^{2} \) |
| 19 | \( 1 + 925 T + p^{5} T^{2} \) |
| 23 | \( 1 - 102 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 726 T + p^{5} T^{2} \) |
| 31 | \( 1 + 811 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7922 T + p^{5} T^{2} \) |
| 41 | \( 1 + 360 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4951 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9906 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8292 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7014 T + p^{5} T^{2} \) |
| 61 | \( 1 + 51433 T + p^{5} T^{2} \) |
| 67 | \( 1 + 581 T + p^{5} T^{2} \) |
| 71 | \( 1 + 56520 T + p^{5} T^{2} \) |
| 73 | \( 1 - 42478 T + p^{5} T^{2} \) |
| 79 | \( 1 + 28912 T + p^{5} T^{2} \) |
| 83 | \( 1 - 104586 T + p^{5} T^{2} \) |
| 89 | \( 1 + 118080 T + p^{5} T^{2} \) |
| 97 | \( 1 + 110273 T + p^{5} T^{2} \) |
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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
−10.47287468576976037467614802264, −9.352596581321208080429437824607, −8.550668640486196449214083912925, −7.58446989469999578597494964774, −6.70949045581553805031685612367, −5.14765083783634346519215414588, −4.34606286253804940725907807882, −2.79653996687212278518158680694, −1.81584845683401866204617240394, 0,
1.81584845683401866204617240394, 2.79653996687212278518158680694, 4.34606286253804940725907807882, 5.14765083783634346519215414588, 6.70949045581553805031685612367, 7.58446989469999578597494964774, 8.550668640486196449214083912925, 9.352596581321208080429437824607, 10.47287468576976037467614802264
