L(s) = 1 | − 4.47·3-s − 5·5-s − 13.4·7-s + 11.0·9-s + 22.3·15-s + 60.0·21-s − 13.4·23-s + 25·25-s − 8.94·27-s + 22·29-s + 67.0·35-s − 62·41-s − 40.2·43-s − 55.0·45-s + 93.9·47-s + 131.·49-s − 58·61-s − 147.·63-s + 67.0·67-s + 60.0·69-s − 111.·75-s − 58.9·81-s − 147.·83-s − 98.3·87-s − 142·89-s − 122·101-s + 201.·103-s + ⋯ |
L(s) = 1 | − 1.49·3-s − 5-s − 1.91·7-s + 1.22·9-s + 1.49·15-s + 2.85·21-s − 0.583·23-s + 25-s − 0.331·27-s + 0.758·29-s + 1.91·35-s − 1.51·41-s − 0.936·43-s − 1.22·45-s + 1.99·47-s + 2.67·49-s − 0.950·61-s − 2.34·63-s + 1.00·67-s + 0.869·69-s − 1.49·75-s − 0.728·81-s − 1.77·83-s − 1.13·87-s − 1.59·89-s − 1.20·101-s + 1.95·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3472063152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3472063152\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 3 | \( 1 + 4.47T + 9T^{2} \) |
| 7 | \( 1 + 13.4T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 - 22T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 62T + 1.68e3T^{2} \) |
| 43 | \( 1 + 40.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 93.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 58T + 3.72e3T^{2} \) |
| 67 | \( 1 - 67.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 147.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 142T + 7.92e3T^{2} \) |
| 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
−11.56117171340888901574378574228, −10.52483875360378107468697209370, −9.885885189694349894111225553592, −8.622109816211932749322263760521, −7.16499615184266096802068037472, −6.53062717079596118228146848895, −5.62572984799509809767393114147, −4.32279713789688332481359824539, −3.18644733930151729217663048696, −0.48446192460622517798641600718,
0.48446192460622517798641600718, 3.18644733930151729217663048696, 4.32279713789688332481359824539, 5.62572984799509809767393114147, 6.53062717079596118228146848895, 7.16499615184266096802068037472, 8.622109816211932749322263760521, 9.885885189694349894111225553592, 10.52483875360378107468697209370, 11.56117171340888901574378574228