| L(s) = 1 | + 0.518·2-s − 1.73·4-s − 4.37·5-s − 5.00·7-s − 1.93·8-s − 2.26·10-s − 1.38·11-s + 3.17·13-s − 2.59·14-s + 2.45·16-s + 4.31·17-s − 1.85·19-s + 7.57·20-s − 0.719·22-s − 5.62·23-s + 14.1·25-s + 1.64·26-s + 8.67·28-s − 5.54·29-s − 6.48·31-s + 5.14·32-s + 2.23·34-s + 21.9·35-s + 7.44·37-s − 0.960·38-s + 8.46·40-s − 3.29·41-s + ⋯ |
| L(s) = 1 | + 0.366·2-s − 0.865·4-s − 1.95·5-s − 1.89·7-s − 0.684·8-s − 0.717·10-s − 0.418·11-s + 0.881·13-s − 0.694·14-s + 0.614·16-s + 1.04·17-s − 0.424·19-s + 1.69·20-s − 0.153·22-s − 1.17·23-s + 2.82·25-s + 0.323·26-s + 1.63·28-s − 1.02·29-s − 1.16·31-s + 0.909·32-s + 0.383·34-s + 3.70·35-s + 1.22·37-s − 0.155·38-s + 1.33·40-s − 0.514·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3872954969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3872954969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.518T + 2T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 + 0.636T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−9.761034472079995230940573106986, −8.928360358578032971949746158065, −8.148967854519683864057571097424, −7.44596460902563604119165848443, −6.39486285215917570940904088126, −5.54318358920765930133982434871, −4.25570960394544152649075039360, −3.60726490652939477086122941258, −3.19747146775704137698863663892, −0.42533630261377754907318204427,
0.42533630261377754907318204427, 3.19747146775704137698863663892, 3.60726490652939477086122941258, 4.25570960394544152649075039360, 5.54318358920765930133982434871, 6.39486285215917570940904088126, 7.44596460902563604119165848443, 8.148967854519683864057571097424, 8.928360358578032971949746158065, 9.761034472079995230940573106986
