| L(s) = 1 | + 2-s + 2.73·3-s + 4-s + 1.56·5-s + 2.73·6-s + 1.91·7-s + 8-s + 4.45·9-s + 1.56·10-s − 0.922·11-s + 2.73·12-s − 3.62·13-s + 1.91·14-s + 4.26·15-s + 16-s + 7.80·17-s + 4.45·18-s + 1.59·19-s + 1.56·20-s + 5.24·21-s − 0.922·22-s + 23-s + 2.73·24-s − 2.56·25-s − 3.62·26-s + 3.97·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.697·5-s + 1.11·6-s + 0.725·7-s + 0.353·8-s + 1.48·9-s + 0.493·10-s − 0.278·11-s + 0.788·12-s − 1.00·13-s + 0.512·14-s + 1.09·15-s + 0.250·16-s + 1.89·17-s + 1.04·18-s + 0.365·19-s + 0.348·20-s + 1.14·21-s − 0.196·22-s + 0.208·23-s + 0.557·24-s − 0.513·25-s − 0.710·26-s + 0.764·27-s + 0.362·28-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
| \(L(1)\) |
\(\approx\) |
\(7.359826086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.359826086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 + 0.922T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 0.736T + 43T^{2} \) |
| 47 | \( 1 - 3.10T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 + 3.31T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + 0.285T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.979870341264577682964947464912, −7.54491119187458108346651847619, −6.80431576383951074910397573962, −5.74435851839209414802430816803, −5.10180261430393415299165080564, −4.44889829196247594971709208994, −3.33283141853189082578215610297, −2.96374804909072119451518972371, −2.05678358809791722973658851669, −1.38291837992765352162801909096,
1.38291837992765352162801909096, 2.05678358809791722973658851669, 2.96374804909072119451518972371, 3.33283141853189082578215610297, 4.44889829196247594971709208994, 5.10180261430393415299165080564, 5.74435851839209414802430816803, 6.80431576383951074910397573962, 7.54491119187458108346651847619, 7.979870341264577682964947464912
