| L(s) = 1 | − 1.79·3-s − 3·5-s + 7-s + 0.208·9-s − 0.791·11-s + 13-s + 5.37·15-s − 0.791·17-s − 19-s − 1.79·21-s − 4.58·23-s + 4·25-s + 5.00·27-s + 0.791·29-s − 6.37·31-s + 1.41·33-s − 3·35-s − 5·37-s − 1.79·39-s + 0.791·41-s − 2·43-s − 0.626·45-s + 1.41·47-s + 49-s + 1.41·51-s − 5.37·53-s + 2.37·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s − 1.34·5-s + 0.377·7-s + 0.0695·9-s − 0.238·11-s + 0.277·13-s + 1.38·15-s − 0.191·17-s − 0.229·19-s − 0.390·21-s − 0.955·23-s + 0.800·25-s + 0.962·27-s + 0.146·29-s − 1.14·31-s + 0.246·33-s − 0.507·35-s − 0.821·37-s − 0.286·39-s + 0.123·41-s − 0.304·43-s − 0.0933·45-s + 0.206·47-s + 0.142·49-s + 0.198·51-s − 0.738·53-s + 0.320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3569412492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3569412492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 0.791T + 17T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 0.791T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 0.791T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 4.37T + 67T^{2} \) |
| 71 | \( 1 - 6.16T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 0.626T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.81504786353188156896870995315, −7.05238747727061286609432930959, −6.44792033160151507544490550853, −5.59535575758865086949932984598, −5.09808561188218344168370245259, −4.22431401813120300732534441531, −3.74797915081579832167505600651, −2.72725899636677242130270388364, −1.53534522516950022267575400039, −0.31281065433610855304088904117,
0.31281065433610855304088904117, 1.53534522516950022267575400039, 2.72725899636677242130270388364, 3.74797915081579832167505600651, 4.22431401813120300732534441531, 5.09808561188218344168370245259, 5.59535575758865086949932984598, 6.44792033160151507544490550853, 7.05238747727061286609432930959, 7.81504786353188156896870995315
