| L(s) = 1 | − 1.32·2-s − 2.26·3-s − 0.242·4-s − 1.07·5-s + 3.00·6-s − 7-s + 2.97·8-s + 2.14·9-s + 1.43·10-s − 4.78·11-s + 0.550·12-s − 3.48·13-s + 1.32·14-s + 2.44·15-s − 3.45·16-s − 2.77·17-s − 2.84·18-s − 2.57·19-s + 0.261·20-s + 2.26·21-s + 6.34·22-s − 0.614·23-s − 6.74·24-s − 3.83·25-s + 4.62·26-s + 1.93·27-s + 0.242·28-s + ⋯ |
| L(s) = 1 | − 0.937·2-s − 1.30·3-s − 0.121·4-s − 0.482·5-s + 1.22·6-s − 0.377·7-s + 1.05·8-s + 0.715·9-s + 0.452·10-s − 1.44·11-s + 0.158·12-s − 0.967·13-s + 0.354·14-s + 0.632·15-s − 0.864·16-s − 0.672·17-s − 0.670·18-s − 0.591·19-s + 0.0585·20-s + 0.495·21-s + 1.35·22-s − 0.128·23-s − 1.37·24-s − 0.766·25-s + 0.906·26-s + 0.372·27-s + 0.0458·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08421608304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08421608304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 0.614T + 23T^{2} \) |
| 29 | \( 1 + 7.90T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 - 8.64T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 0.747T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 0.421T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 7.71T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−10.19346927494926050445868217858, −9.449931954605002063461630378982, −8.430319993266971449690677480607, −7.58358141733794173303140657292, −6.93520732438140539866309152678, −5.66103262141820092340255621535, −5.01464627770258114678156576108, −3.99709867019463552696064578800, −2.22535743645671372498604603655, −0.26739432089065209146931607778,
0.26739432089065209146931607778, 2.22535743645671372498604603655, 3.99709867019463552696064578800, 5.01464627770258114678156576108, 5.66103262141820092340255621535, 6.93520732438140539866309152678, 7.58358141733794173303140657292, 8.430319993266971449690677480607, 9.449931954605002063461630378982, 10.19346927494926050445868217858
