L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s − 12-s − 14-s + 2·15-s + 16-s + 17-s − 18-s + 6·19-s − 2·20-s − 21-s + 3·23-s + 24-s − 25-s − 27-s + 28-s + 6·29-s − 2·30-s + 5·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 0.218·21-s + 0.625·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.365·30-s + 0.898·31-s − 0.176·32-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
\(L(1)\) |
\(\approx\) |
\(0.9893716274\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9893716274\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.918513156543623948981841462025, −7.29328873971770510713284329806, −6.79910881895436687797766766060, −5.85936116148070757682859528303, −5.15485033558673190802567841947, −4.41638654933839715882264674687, −3.51091376550393319678388627849, −2.70622957309848755649934307908, −1.42869996178933686836232161782, −0.63157255227202246148724167428,
0.63157255227202246148724167428, 1.42869996178933686836232161782, 2.70622957309848755649934307908, 3.51091376550393319678388627849, 4.41638654933839715882264674687, 5.15485033558673190802567841947, 5.85936116148070757682859528303, 6.79910881895436687797766766060, 7.29328873971770510713284329806, 7.918513156543623948981841462025