L(s) = 1 | − 10.8·7-s + 9·9-s − 17.3·11-s + 2.26·17-s + 19·19-s − 34.8·23-s + 80.1·43-s − 93.2·47-s + 69.1·49-s + 108.·61-s − 97.8·63-s + 50.2·73-s + 188.·77-s + 81·81-s + 139.·83-s − 156.·99-s + 102·101-s − 24.6·119-s + ⋯ |
L(s) = 1 | − 1.55·7-s + 9-s − 1.57·11-s + 0.133·17-s + 19-s − 1.51·23-s + 1.86·43-s − 1.98·47-s + 1.41·49-s + 1.77·61-s − 1.55·63-s + 0.688·73-s + 2.45·77-s + 81-s + 1.68·83-s − 1.57·99-s + 1.00·101-s − 0.206·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.238994303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238994303\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
good | 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 + 10.8T + 49T^{2} \) |
| 11 | \( 1 + 17.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 2.26T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 80.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 93.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.263480726024719081869909800032, −8.058397664386923423743556184693, −7.50507171994877416923070230825, −6.66847548424731724854005117235, −5.87953670778301954870094096230, −5.04375841878406983857268005986, −3.93532535501256035944749171463, −3.13256354640603480040408029735, −2.16183517418428636775753879775, −0.57662837805838410750133109559,
0.57662837805838410750133109559, 2.16183517418428636775753879775, 3.13256354640603480040408029735, 3.93532535501256035944749171463, 5.04375841878406983857268005986, 5.87953670778301954870094096230, 6.66847548424731724854005117235, 7.50507171994877416923070230825, 8.058397664386923423743556184693, 9.263480726024719081869909800032