L(s) = 1 | − 2-s + 0.309·3-s + 4-s − 0.0978·5-s − 0.309·6-s + 3.92·7-s − 8-s − 2.90·9-s + 0.0978·10-s − 5.36·11-s + 0.309·12-s + 3.25·13-s − 3.92·14-s − 0.0302·15-s + 16-s + 2.13·17-s + 2.90·18-s − 1.65·19-s − 0.0978·20-s + 1.21·21-s + 5.36·22-s − 5.57·23-s − 0.309·24-s − 4.99·25-s − 3.25·26-s − 1.82·27-s + 3.92·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.178·3-s + 0.5·4-s − 0.0437·5-s − 0.126·6-s + 1.48·7-s − 0.353·8-s − 0.968·9-s + 0.0309·10-s − 1.61·11-s + 0.0892·12-s + 0.904·13-s − 1.04·14-s − 0.00780·15-s + 0.250·16-s + 0.517·17-s + 0.684·18-s − 0.379·19-s − 0.0218·20-s + 0.264·21-s + 1.14·22-s − 1.16·23-s − 0.0631·24-s − 0.998·25-s − 0.639·26-s − 0.351·27-s + 0.741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.366121724\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.366121724\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 0.309T + 3T^{2} \) |
| 5 | \( 1 + 0.0978T + 5T^{2} \) |
| 7 | \( 1 - 3.92T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 + 2.28T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 + 0.328T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 + 6.69T + 97T^{2} \) |
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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
−8.192091731919339974196230891378, −8.010759999400624335315724979145, −7.43497321019282531634178296771, −5.97421994782061794555304386466, −5.73960503046568516788277925110, −4.74497402428988259067105950327, −3.78455049929945193037503889774, −2.59604096635389353231281688484, −2.03593022085319539751242016737, −0.72563063540636428351238176730,
0.72563063540636428351238176730, 2.03593022085319539751242016737, 2.59604096635389353231281688484, 3.78455049929945193037503889774, 4.74497402428988259067105950327, 5.73960503046568516788277925110, 5.97421994782061794555304386466, 7.43497321019282531634178296771, 8.010759999400624335315724979145, 8.192091731919339974196230891378