| L(s) = 1 | + 0.208·7-s + 11-s + 13-s + 0.791·17-s + 6.58·19-s − 3.79·23-s − 6.79·29-s − 8.58·31-s + 2.58·37-s + 1.41·41-s + 10·43-s − 1.41·47-s − 6.95·49-s + 11.3·53-s + 10.5·59-s + 4.20·61-s + 4·67-s + 10.7·71-s + 7.79·73-s + 0.208·77-s − 15.5·79-s − 9.95·83-s + 0.791·89-s + 0.208·91-s + 6.20·97-s + 17.3·101-s − 5.95·103-s + ⋯ |
| L(s) = 1 | + 0.0788·7-s + 0.301·11-s + 0.277·13-s + 0.191·17-s + 1.51·19-s − 0.790·23-s − 1.26·29-s − 1.54·31-s + 0.424·37-s + 0.221·41-s + 1.52·43-s − 0.206·47-s − 0.993·49-s + 1.56·53-s + 1.37·59-s + 0.538·61-s + 0.488·67-s + 1.27·71-s + 0.911·73-s + 0.0237·77-s − 1.74·79-s − 1.09·83-s + 0.0838·89-s + 0.0218·91-s + 0.630·97-s + 1.72·101-s − 0.586·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.149190246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149190246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 0.208T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.791T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.20T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 - 0.791T + 89T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−7.43432781984168808114136997720, −7.26506315697794463932998477885, −6.19243485647893346799601012734, −5.60647051830242820380083084977, −5.07407726911444998021418720833, −3.92704240708995900187317252822, −3.64077657863934102635932204261, −2.55736291313798276264405533976, −1.70022781428777716407173640541, −0.71157177702704428204962183639,
0.71157177702704428204962183639, 1.70022781428777716407173640541, 2.55736291313798276264405533976, 3.64077657863934102635932204261, 3.92704240708995900187317252822, 5.07407726911444998021418720833, 5.60647051830242820380083084977, 6.19243485647893346799601012734, 7.26506315697794463932998477885, 7.43432781984168808114136997720
