| L(s) = 1 | + 0.475·2-s + 0.345·3-s − 1.77·4-s + 0.161·5-s + 0.164·6-s − 4.07·7-s − 1.79·8-s − 2.88·9-s + 0.0768·10-s − 3.77·11-s − 0.613·12-s − 1.94·14-s + 0.0558·15-s + 2.69·16-s − 1.83·17-s − 1.37·18-s − 6.07·19-s − 0.286·20-s − 1.41·21-s − 1.79·22-s − 8.58·23-s − 0.620·24-s − 4.97·25-s − 2.03·27-s + 7.23·28-s − 5.21·29-s + 0.0265·30-s + ⋯ |
| L(s) = 1 | + 0.336·2-s + 0.199·3-s − 0.886·4-s + 0.0722·5-s + 0.0671·6-s − 1.54·7-s − 0.634·8-s − 0.960·9-s + 0.0243·10-s − 1.13·11-s − 0.176·12-s − 0.518·14-s + 0.0144·15-s + 0.673·16-s − 0.444·17-s − 0.322·18-s − 1.39·19-s − 0.0640·20-s − 0.307·21-s − 0.383·22-s − 1.79·23-s − 0.126·24-s − 0.994·25-s − 0.391·27-s + 1.36·28-s − 0.968·29-s + 0.00485·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06079947255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06079947255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.475T + 2T^{2} \) |
| 3 | \( 1 - 0.345T + 3T^{2} \) |
| 5 | \( 1 - 0.161T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 7.69T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 + 0.366T + 97T^{2} \) |
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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
−8.124462460136910534553369408216, −7.75886768765522708979617185966, −6.44495257116442927199703830059, −5.95493848288853288986595252965, −5.49543111110823113654540907937, −4.29083202023637872576985261150, −3.82781167486319229100950093207, −2.87889134053674048647633427893, −2.26298283344592729070917602250, −0.11462791971779993793980432680,
0.11462791971779993793980432680, 2.26298283344592729070917602250, 2.87889134053674048647633427893, 3.82781167486319229100950093207, 4.29083202023637872576985261150, 5.49543111110823113654540907937, 5.95493848288853288986595252965, 6.44495257116442927199703830059, 7.75886768765522708979617185966, 8.124462460136910534553369408216
