| L(s) = 1 | + 2·2-s − 7.73·3-s + 4·4-s − 15.4·6-s − 23.5·7-s + 8·8-s + 32.7·9-s − 32.1·11-s − 30.9·12-s − 40.0·13-s − 47.1·14-s + 16·16-s − 126.·17-s + 65.5·18-s + 0.232·19-s + 182.·21-s − 64.2·22-s + 23·23-s − 61.8·24-s − 80.1·26-s − 44.5·27-s − 94.2·28-s − 137.·29-s + 112.·31-s + 32·32-s + 248.·33-s − 252.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.48·3-s + 0.5·4-s − 1.05·6-s − 1.27·7-s + 0.353·8-s + 1.21·9-s − 0.880·11-s − 0.743·12-s − 0.855·13-s − 0.899·14-s + 0.250·16-s − 1.79·17-s + 0.858·18-s + 0.00281·19-s + 1.89·21-s − 0.622·22-s + 0.208·23-s − 0.526·24-s − 0.604·26-s − 0.317·27-s − 0.636·28-s − 0.878·29-s + 0.653·31-s + 0.176·32-s + 1.31·33-s − 1.27·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4723011116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4723011116\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 7.73T + 27T^{2} \) |
| 7 | \( 1 + 23.5T + 343T^{2} \) |
| 11 | \( 1 + 32.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.232T + 6.85e3T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 45.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 543.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 26.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 43.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 202.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 150.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 667.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 602.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 485.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.851042190140741613222995118648, −8.622890586126223640063294186009, −7.23991003226281376966233414160, −6.68981266677302054500238558850, −6.03164704783195438591295746694, −5.14321967409647888074247283958, −4.54008364859338653568598130767, −3.27671817268913015900782136382, −2.16555958730734190863171700395, −0.32015779495147579307099293244,
0.32015779495147579307099293244, 2.16555958730734190863171700395, 3.27671817268913015900782136382, 4.54008364859338653568598130767, 5.14321967409647888074247283958, 6.03164704783195438591295746694, 6.68981266677302054500238558850, 7.23991003226281376966233414160, 8.622890586126223640063294186009, 9.851042190140741613222995118648
