L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 3·7-s + 9-s − 2·10-s − 3·11-s + 2·12-s − 6·13-s − 6·14-s + 15-s − 4·16-s + 3·17-s − 2·18-s − 19-s + 2·20-s + 3·21-s + 6·22-s + 4·23-s − 4·25-s + 12·26-s + 27-s + 6·28-s − 10·29-s − 2·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s + 0.577·12-s − 1.66·13-s − 1.60·14-s + 0.258·15-s − 16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s + 1.27·22-s + 0.834·23-s − 4/5·25-s + 2.35·26-s + 0.192·27-s + 1.13·28-s − 1.85·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5865095916\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5865095916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.26425654515921203605388116110, −14.35869809327674691065038767453, −13.04832504304323016422711730155, −11.43002930912671853728259843597, −10.20230313909289126093125054009, −9.400588261603299107064903686230, −8.062798749262723709584757589645, −7.42319981131404835915630105083, −5.02993077790447449830764321556, −2.13307257724406321246824925475,
2.13307257724406321246824925475, 5.02993077790447449830764321556, 7.42319981131404835915630105083, 8.062798749262723709584757589645, 9.400588261603299107064903686230, 10.20230313909289126093125054009, 11.43002930912671853728259843597, 13.04832504304323016422711730155, 14.35869809327674691065038767453, 15.26425654515921203605388116110