| L(s) = 1 | − 2-s − 1.80·3-s + 4-s − 1.32·5-s + 1.80·6-s − 4.32·7-s − 8-s + 0.260·9-s + 1.32·10-s + 1.86·11-s − 1.80·12-s + 3.09·13-s + 4.32·14-s + 2.39·15-s + 16-s − 4.60·17-s − 0.260·18-s − 5.39·19-s − 1.32·20-s + 7.81·21-s − 1.86·22-s + 23-s + 1.80·24-s − 3.23·25-s − 3.09·26-s + 4.94·27-s − 4.32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s − 0.593·5-s + 0.737·6-s − 1.63·7-s − 0.353·8-s + 0.0867·9-s + 0.419·10-s + 0.562·11-s − 0.521·12-s + 0.857·13-s + 1.15·14-s + 0.618·15-s + 0.250·16-s − 1.11·17-s − 0.0613·18-s − 1.23·19-s − 0.296·20-s + 1.70·21-s − 0.397·22-s + 0.208·23-s + 0.368·24-s − 0.647·25-s − 0.606·26-s + 0.952·27-s − 0.817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01552503974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01552503974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 5.39T + 19T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 + 2.10T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 - 5.88T + 61T^{2} \) |
| 67 | \( 1 + 1.51T + 67T^{2} \) |
| 71 | \( 1 + 6.94T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 8.12T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.312701325118569396588432058301, −6.97312239137657533270480212500, −6.77102688107882840211807662591, −6.14687373084101858831273151902, −5.53145564050972445347065879917, −4.30972311276275723904338608541, −3.66644861446861042540742083127, −2.79730581091489976018718607767, −1.54538725197243334620487329655, −0.07627801084746191496414653722,
0.07627801084746191496414653722, 1.54538725197243334620487329655, 2.79730581091489976018718607767, 3.66644861446861042540742083127, 4.30972311276275723904338608541, 5.53145564050972445347065879917, 6.14687373084101858831273151902, 6.77102688107882840211807662591, 6.97312239137657533270480212500, 8.312701325118569396588432058301
