| L(s) = 1 | − 1.18·2-s − 0.604·4-s + 2.90·5-s − 0.528·7-s + 3.07·8-s − 3.42·10-s + 11-s − 0.150·13-s + 0.624·14-s − 2.42·16-s − 0.0673·17-s − 0.482·19-s − 1.75·20-s − 1.18·22-s + 4.10·23-s + 3.41·25-s + 0.178·26-s + 0.319·28-s + 5.83·29-s − 2.84·31-s − 3.28·32-s + 0.0795·34-s − 1.53·35-s + 6.02·37-s + 0.569·38-s + 8.92·40-s − 11.4·41-s + ⋯ |
| L(s) = 1 | − 0.835·2-s − 0.302·4-s + 1.29·5-s − 0.199·7-s + 1.08·8-s − 1.08·10-s + 0.301·11-s − 0.0418·13-s + 0.166·14-s − 0.606·16-s − 0.0163·17-s − 0.110·19-s − 0.391·20-s − 0.251·22-s + 0.856·23-s + 0.682·25-s + 0.0349·26-s + 0.0603·28-s + 1.08·29-s − 0.510·31-s − 0.581·32-s + 0.0136·34-s − 0.259·35-s + 0.991·37-s + 0.0924·38-s + 1.41·40-s − 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473229025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473229025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 + 0.528T + 7T^{2} \) |
| 13 | \( 1 + 0.150T + 13T^{2} \) |
| 17 | \( 1 + 0.0673T + 17T^{2} \) |
| 19 | \( 1 + 0.482T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 6.02T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 0.588T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.221721489446094244262803625259, −7.48538470267631084759690213103, −6.65627845657887208633866651495, −6.08646457178597433760199412432, −5.18461188857912481051671336487, −4.62373633007760209266167763218, −3.57500623869202196276361111457, −2.51289218475540666450587825300, −1.65322550260515605774620357453, −0.76646874793059205121306802660,
0.76646874793059205121306802660, 1.65322550260515605774620357453, 2.51289218475540666450587825300, 3.57500623869202196276361111457, 4.62373633007760209266167763218, 5.18461188857912481051671336487, 6.08646457178597433760199412432, 6.65627845657887208633866651495, 7.48538470267631084759690213103, 8.221721489446094244262803625259
