L(s) = 1 | − 1.83·2-s − 1.64·3-s + 1.38·4-s + 3.02·6-s + 7-s + 1.13·8-s − 0.291·9-s − 2.13·11-s − 2.27·12-s + 3.30·13-s − 1.83·14-s − 4.85·16-s + 4.46·17-s + 0.536·18-s + 4.87·19-s − 1.64·21-s + 3.92·22-s − 23-s − 1.86·24-s − 6.08·26-s + 5.41·27-s + 1.38·28-s − 7.83·29-s + 10.2·31-s + 6.65·32-s + 3.51·33-s − 8.22·34-s + ⋯ |
L(s) = 1 | − 1.30·2-s − 0.950·3-s + 0.691·4-s + 1.23·6-s + 0.377·7-s + 0.400·8-s − 0.0972·9-s − 0.644·11-s − 0.657·12-s + 0.916·13-s − 0.491·14-s − 1.21·16-s + 1.08·17-s + 0.126·18-s + 1.11·19-s − 0.359·21-s + 0.837·22-s − 0.208·23-s − 0.380·24-s − 1.19·26-s + 1.04·27-s + 0.261·28-s − 1.45·29-s + 1.83·31-s + 1.17·32-s + 0.611·33-s − 1.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6336133980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6336133980\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.176T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 8.95T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.368281102722730857757469873163, −7.87023974449675590602939607848, −7.22397797148732327260497659500, −6.27329595857895479892662642472, −5.53392616118109498163342067242, −4.96855149028398681972238376327, −3.84984020370338680794070928958, −2.70126959745144729030530600132, −1.41070721062660226844214547129, −0.64808613478401777608652392587,
0.64808613478401777608652392587, 1.41070721062660226844214547129, 2.70126959745144729030530600132, 3.84984020370338680794070928958, 4.96855149028398681972238376327, 5.53392616118109498163342067242, 6.27329595857895479892662642472, 7.22397797148732327260497659500, 7.87023974449675590602939607848, 8.368281102722730857757469873163