L(s) = 1 | − 2-s − 3-s + 4-s + 2.29·5-s + 6-s − 0.955·7-s − 8-s + 9-s − 2.29·10-s − 11-s − 12-s + 2.11·13-s + 0.955·14-s − 2.29·15-s + 16-s + 4.97·17-s − 18-s + 7.95·19-s + 2.29·20-s + 0.955·21-s + 22-s + 7.11·23-s + 24-s + 0.245·25-s − 2.11·26-s − 27-s − 0.955·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.02·5-s + 0.408·6-s − 0.361·7-s − 0.353·8-s + 0.333·9-s − 0.724·10-s − 0.301·11-s − 0.288·12-s + 0.587·13-s + 0.255·14-s − 0.591·15-s + 0.250·16-s + 1.20·17-s − 0.235·18-s + 1.82·19-s + 0.512·20-s + 0.208·21-s + 0.213·22-s + 1.48·23-s + 0.204·24-s + 0.0491·25-s − 0.415·26-s − 0.192·27-s − 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.494144942\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494144942\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 2.29T + 5T^{2} \) |
| 7 | \( 1 + 0.955T + 7T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 + 5.59T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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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
−8.546818911130746198846831326159, −7.61433395149371278203118891737, −7.03449716208310755819945738123, −6.23680766177586071079101341793, −5.48779216716465522925843645436, −5.14735702990520573470259947352, −3.59001276950887594637915879666, −2.88151816987193395011279470853, −1.63826782330686292984803819615, −0.861415608405643290772232962048,
0.861415608405643290772232962048, 1.63826782330686292984803819615, 2.88151816987193395011279470853, 3.59001276950887594637915879666, 5.14735702990520573470259947352, 5.48779216716465522925843645436, 6.23680766177586071079101341793, 7.03449716208310755819945738123, 7.61433395149371278203118891737, 8.546818911130746198846831326159