| L(s) = 1 | − 0.532·3-s − 0.120·5-s − 2.71·9-s − 11-s + 1.22·13-s + 0.0641·15-s − 6.17·17-s + 6.41·19-s + 2.02·23-s − 4.98·25-s + 3.04·27-s + 3.24·29-s + 4.87·31-s + 0.532·33-s + 2.36·37-s − 0.652·39-s − 8.29·41-s − 2.22·43-s + 0.327·45-s + 9.23·47-s + 3.28·51-s + 9.68·53-s + 0.120·55-s − 3.41·57-s − 9.74·59-s + 1.43·61-s − 0.147·65-s + ⋯ |
| L(s) = 1 | − 0.307·3-s − 0.0539·5-s − 0.905·9-s − 0.301·11-s + 0.340·13-s + 0.0165·15-s − 1.49·17-s + 1.47·19-s + 0.421·23-s − 0.997·25-s + 0.585·27-s + 0.603·29-s + 0.876·31-s + 0.0926·33-s + 0.389·37-s − 0.104·39-s − 1.29·41-s − 0.339·43-s + 0.0488·45-s + 1.34·47-s + 0.459·51-s + 1.32·53-s + 0.0162·55-s − 0.451·57-s − 1.26·59-s + 0.183·61-s − 0.0183·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.532T + 3T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 9.68T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 - 6.87T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.37260181213630336375899863286, −6.70940037371626646847182253505, −6.02055996521179366297669455901, −5.38477592289239904754605883144, −4.73398356915304203045934025999, −3.88168549533056485701576615027, −2.98693805476551826785848254083, −2.34201440113464566292603724995, −1.11562338447763612182076696248, 0,
1.11562338447763612182076696248, 2.34201440113464566292603724995, 2.98693805476551826785848254083, 3.88168549533056485701576615027, 4.73398356915304203045934025999, 5.38477592289239904754605883144, 6.02055996521179366297669455901, 6.70940037371626646847182253505, 7.37260181213630336375899863286
