| L(s) = 1 | + 5.61·5-s + 8.82·7-s + 13.4·11-s − 2.26·17-s + 19·19-s − 34.8·23-s + 6.52·25-s + 49.5·35-s + 31.1·43-s − 93.2·47-s + 28.8·49-s + 75.7·55-s + 108.·61-s + 137.·73-s + 119.·77-s − 139.·83-s − 12.7·85-s + 106.·95-s + 174.·101-s − 195.·115-s − 19.9·119-s + ⋯ |
| L(s) = 1 | + 1.12·5-s + 1.26·7-s + 1.22·11-s − 0.133·17-s + 19-s − 1.51·23-s + 0.261·25-s + 1.41·35-s + 0.725·43-s − 1.98·47-s + 0.589·49-s + 1.37·55-s + 1.77·61-s + 1.87·73-s + 1.54·77-s − 1.68·83-s − 0.149·85-s + 1.12·95-s + 1.72·101-s − 1.70·115-s − 0.168·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.780956638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.780956638\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 5.61T + 25T^{2} \) |
| 7 | \( 1 - 8.82T + 49T^{2} \) |
| 11 | \( 1 - 13.4T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 2.26T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 93.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.07720673722218435291283092148, −9.547692252164941430876700599513, −8.585386464075352228575737332960, −7.74356796732597718302830012140, −6.61333178367685072636144820323, −5.77197329402580987089122235561, −4.88616679476817671827706909345, −3.77808947308780840039653908236, −2.15802858015011876427449359354, −1.31846967635821519710982968441,
1.31846967635821519710982968441, 2.15802858015011876427449359354, 3.77808947308780840039653908236, 4.88616679476817671827706909345, 5.77197329402580987089122235561, 6.61333178367685072636144820323, 7.74356796732597718302830012140, 8.585386464075352228575737332960, 9.547692252164941430876700599513, 10.07720673722218435291283092148
