| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s + 5.17·11-s − 12-s − 14-s − 2·15-s + 16-s + 5·17-s − 18-s − 3.17·19-s + 2·20-s − 21-s − 5.17·22-s − 8.17·23-s + 24-s − 25-s − 27-s + 28-s − 3.17·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.632·10-s + 1.56·11-s − 0.288·12-s − 0.267·14-s − 0.516·15-s + 0.250·16-s + 1.21·17-s − 0.235·18-s − 0.729·19-s + 0.447·20-s − 0.218·21-s − 1.10·22-s − 1.70·23-s + 0.204·24-s − 0.200·25-s − 0.192·27-s + 0.188·28-s − 0.590·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664796518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664796518\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7.17T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 3.82T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.83676740257569031736813294144, −7.33133297209796264462224475836, −6.40921079103539980944260610478, −5.92702329900097779158484049030, −5.48475911813519175659179726696, −4.22643559376834830035243990149, −3.71720204573071596317679520498, −2.27941211470107522943892843184, −1.68571350499056331620649629374, −0.78819259466702459849253676515,
0.78819259466702459849253676515, 1.68571350499056331620649629374, 2.27941211470107522943892843184, 3.71720204573071596317679520498, 4.22643559376834830035243990149, 5.48475911813519175659179726696, 5.92702329900097779158484049030, 6.40921079103539980944260610478, 7.33133297209796264462224475836, 7.83676740257569031736813294144
