| L(s) = 1 | − 0.720·2-s − 0.439·3-s − 1.48·4-s + 0.316·6-s + 1.55·7-s + 2.50·8-s − 2.80·9-s + 1.48·11-s + 0.650·12-s − 5.11·13-s − 1.12·14-s + 1.15·16-s + 4.91·17-s + 2.02·18-s + 2.65·19-s − 0.683·21-s − 1.07·22-s − 7.51·23-s − 1.10·24-s + 3.68·26-s + 2.55·27-s − 2.30·28-s + 7.73·29-s + 1.77·31-s − 5.84·32-s − 0.653·33-s − 3.54·34-s + ⋯ |
| L(s) = 1 | − 0.509·2-s − 0.253·3-s − 0.740·4-s + 0.129·6-s + 0.588·7-s + 0.886·8-s − 0.935·9-s + 0.448·11-s + 0.187·12-s − 1.41·13-s − 0.299·14-s + 0.288·16-s + 1.19·17-s + 0.476·18-s + 0.609·19-s − 0.149·21-s − 0.228·22-s − 1.56·23-s − 0.224·24-s + 0.722·26-s + 0.490·27-s − 0.435·28-s + 1.43·29-s + 0.319·31-s − 1.03·32-s − 0.113·33-s − 0.607·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.720T + 2T^{2} \) |
| 3 | \( 1 + 0.439T + 3T^{2} \) |
| 7 | \( 1 - 1.55T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 + 0.817T + 47T^{2} \) |
| 53 | \( 1 + 7.10T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 67 | \( 1 + 4.18T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 0.305T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 4.51T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−9.067083187427873364856507956426, −8.184724443638109049878396513663, −7.81759182761751828336412346575, −6.73508595251459670752753674026, −5.52797520717505334301583976433, −5.03995311840705963553124622933, −4.05502017854191262034648980291, −2.85133950411320020357270911738, −1.41924477879589327092257073348, 0,
1.41924477879589327092257073348, 2.85133950411320020357270911738, 4.05502017854191262034648980291, 5.03995311840705963553124622933, 5.52797520717505334301583976433, 6.73508595251459670752753674026, 7.81759182761751828336412346575, 8.184724443638109049878396513663, 9.067083187427873364856507956426
