| L(s) = 1 | − 0.590·2-s + 0.738·3-s − 1.65·4-s − 5-s − 0.436·6-s − 1.02·7-s + 2.15·8-s − 2.45·9-s + 0.590·10-s + 11-s − 1.21·12-s − 3.73·13-s + 0.606·14-s − 0.738·15-s + 2.02·16-s − 0.819·17-s + 1.44·18-s + 6.13·19-s + 1.65·20-s − 0.758·21-s − 0.590·22-s + 3.31·23-s + 1.59·24-s + 25-s + 2.20·26-s − 4.02·27-s + 1.69·28-s + ⋯ |
| L(s) = 1 | − 0.417·2-s + 0.426·3-s − 0.825·4-s − 0.447·5-s − 0.178·6-s − 0.388·7-s + 0.762·8-s − 0.818·9-s + 0.186·10-s + 0.301·11-s − 0.351·12-s − 1.03·13-s + 0.162·14-s − 0.190·15-s + 0.507·16-s − 0.198·17-s + 0.341·18-s + 1.40·19-s + 0.369·20-s − 0.165·21-s − 0.125·22-s + 0.692·23-s + 0.325·24-s + 0.200·25-s + 0.432·26-s − 0.775·27-s + 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7391839563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7391839563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.590T + 2T^{2} \) |
| 3 | \( 1 - 0.738T + 3T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 + 0.819T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 + 0.856T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.357T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 0.314T + 43T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 1.74T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 6.85T + 67T^{2} \) |
| 71 | \( 1 - 2.28T + 71T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.577825509835561686166714883774, −7.71703296729590576055075375892, −7.35210985718422332269983729593, −6.31202610942578079008046646827, −5.18990925215333213565557992200, −4.85664064500729179843087371612, −3.55373209182640980998834727654, −3.22755571086812914204398487258, −1.90018683430408143675004898336, −0.50544587211578523915873460551,
0.50544587211578523915873460551, 1.90018683430408143675004898336, 3.22755571086812914204398487258, 3.55373209182640980998834727654, 4.85664064500729179843087371612, 5.18990925215333213565557992200, 6.31202610942578079008046646827, 7.35210985718422332269983729593, 7.71703296729590576055075375892, 8.577825509835561686166714883774
