| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3.12·7-s − 8-s + 9-s − 3.12·11-s + 12-s − 2·13-s + 3.12·14-s + 16-s + 1.12·17-s − 18-s + 4·19-s − 3.12·21-s + 3.12·22-s − 23-s − 24-s + 2·26-s + 27-s − 3.12·28-s + 2·29-s − 32-s − 3.12·33-s − 1.12·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.18·7-s − 0.353·8-s + 0.333·9-s − 0.941·11-s + 0.288·12-s − 0.554·13-s + 0.834·14-s + 0.250·16-s + 0.272·17-s − 0.235·18-s + 0.917·19-s − 0.681·21-s + 0.665·22-s − 0.208·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.590·28-s + 0.371·29-s − 0.176·32-s − 0.543·33-s − 0.192·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138122157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138122157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.646679262043882079660350934941, −7.81088795689132586530768703778, −7.34914820501121458376115353228, −6.54617218469250239751285841744, −5.72116088804945613648435604920, −4.81152972952805158930851294096, −3.55756712798382439185815253945, −2.93856716003623527264808911580, −2.12528285019996733619865613515, −0.65534543893348699897491693365,
0.65534543893348699897491693365, 2.12528285019996733619865613515, 2.93856716003623527264808911580, 3.55756712798382439185815253945, 4.81152972952805158930851294096, 5.72116088804945613648435604920, 6.54617218469250239751285841744, 7.34914820501121458376115353228, 7.81088795689132586530768703778, 8.646679262043882079660350934941
