L(s) = 1 | − 0.709·2-s − 2.87·3-s − 1.49·4-s − 2.11·5-s + 2.03·6-s − 0.783·7-s + 2.47·8-s + 5.25·9-s + 1.49·10-s − 1.64·11-s + 4.30·12-s + 0.555·14-s + 6.07·15-s + 1.23·16-s − 2.10·17-s − 3.72·18-s − 1.79·19-s + 3.16·20-s + 2.25·21-s + 1.16·22-s − 4.11·23-s − 7.12·24-s − 0.533·25-s − 6.48·27-s + 1.17·28-s − 8.45·29-s − 4.30·30-s + ⋯ |
L(s) = 1 | − 0.501·2-s − 1.65·3-s − 0.748·4-s − 0.945·5-s + 0.831·6-s − 0.296·7-s + 0.876·8-s + 1.75·9-s + 0.473·10-s − 0.497·11-s + 1.24·12-s + 0.148·14-s + 1.56·15-s + 0.309·16-s − 0.509·17-s − 0.878·18-s − 0.410·19-s + 0.707·20-s + 0.491·21-s + 0.249·22-s − 0.858·23-s − 1.45·24-s − 0.106·25-s − 1.24·27-s + 0.221·28-s − 1.57·29-s − 0.786·30-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 0.709T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 0.783T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 + 1.79T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 37 | \( 1 + 0.960T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 + 4.39T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 - 4.65T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 6.55T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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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
−7.82796627009523651892092895146, −7.16581698775528834536829655176, −6.41052816342847201222108982051, −5.54007994587795620271163783225, −5.06544503174861679053705841977, −4.11914155347800331107757434249, −3.77768889356996684055429834759, −2.04683832341263001750115298832, −0.68516933987912741700441627328, 0,
0.68516933987912741700441627328, 2.04683832341263001750115298832, 3.77768889356996684055429834759, 4.11914155347800331107757434249, 5.06544503174861679053705841977, 5.54007994587795620271163783225, 6.41052816342847201222108982051, 7.16581698775528834536829655176, 7.82796627009523651892092895146