| L(s) = 1 | + 2.56·2-s + 1.67·3-s + 4.55·4-s + 5-s + 4.28·6-s + 6.54·8-s − 0.200·9-s + 2.56·10-s + 6.55·11-s + 7.62·12-s + 1.87·13-s + 1.67·15-s + 7.65·16-s − 0.225·17-s − 0.514·18-s + 1.89·19-s + 4.55·20-s + 16.7·22-s − 4.67·23-s + 10.9·24-s + 25-s + 4.79·26-s − 5.35·27-s + 29-s + 4.28·30-s − 7.51·31-s + 6.50·32-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.965·3-s + 2.27·4-s + 0.447·5-s + 1.74·6-s + 2.31·8-s − 0.0669·9-s + 0.809·10-s + 1.97·11-s + 2.20·12-s + 0.518·13-s + 0.431·15-s + 1.91·16-s − 0.0547·17-s − 0.121·18-s + 0.435·19-s + 1.01·20-s + 3.57·22-s − 0.975·23-s + 2.23·24-s + 0.200·25-s + 0.939·26-s − 1.03·27-s + 0.185·29-s + 0.782·30-s − 1.34·31-s + 1.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.15872859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.15872859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 11 | \( 1 - 6.55T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 0.225T + 17T^{2} \) |
| 19 | \( 1 - 1.89T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 2.35T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 0.0442T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 8.84T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 4.55T + 97T^{2} \) |
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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
−7.70623073434986195379626085144, −6.98134429072674037144927075490, −6.28588509486321075428170040510, −5.86688300641147929443467037954, −5.03025015637055421438958415425, −4.04578925327157681606952041255, −3.71085349842584498609074293508, −3.04788218681817572560752759823, −2.08731343509809489183113464360, −1.47986574811676139044695612222,
1.47986574811676139044695612222, 2.08731343509809489183113464360, 3.04788218681817572560752759823, 3.71085349842584498609074293508, 4.04578925327157681606952041255, 5.03025015637055421438958415425, 5.86688300641147929443467037954, 6.28588509486321075428170040510, 6.98134429072674037144927075490, 7.70623073434986195379626085144
