| L(s) = 1 | − 2.61·3-s − 3.61·5-s + 3.85·9-s − 1.23·13-s + 9.47·15-s − 17-s − 6.47·19-s + 0.763·23-s + 8.09·25-s − 2.23·27-s + 7.23·29-s + 10.5·31-s − 9.70·37-s + 3.23·39-s + 7.85·41-s − 1.85·43-s − 13.9·45-s + 6.94·47-s + 2.61·51-s − 3.32·53-s + 16.9·57-s − 0.763·59-s + 7.56·61-s + 4.47·65-s + 6.09·67-s − 2·69-s − 9.23·71-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 1.61·5-s + 1.28·9-s − 0.342·13-s + 2.44·15-s − 0.242·17-s − 1.48·19-s + 0.159·23-s + 1.61·25-s − 0.430·27-s + 1.34·29-s + 1.89·31-s − 1.59·37-s + 0.518·39-s + 1.22·41-s − 0.282·43-s − 2.07·45-s + 1.01·47-s + 0.366·51-s − 0.456·53-s + 2.24·57-s − 0.0994·59-s + 0.968·61-s + 0.554·65-s + 0.744·67-s − 0.240·69-s − 1.09·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 - 7.85T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 3.32T + 53T^{2} \) |
| 59 | \( 1 + 0.763T + 59T^{2} \) |
| 61 | \( 1 - 7.56T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 + 3.85T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.249094780469195592028243824111, −7.35623950739432264766386568065, −6.67668351713048585785681933165, −6.15057464773386584460006655295, −4.98642304413560119702008758584, −4.54330346094629470107763660239, −3.82763275488386918786053268894, −2.60205011479295282947528503767, −0.921674295272300449676830037175, 0,
0.921674295272300449676830037175, 2.60205011479295282947528503767, 3.82763275488386918786053268894, 4.54330346094629470107763660239, 4.98642304413560119702008758584, 6.15057464773386584460006655295, 6.67668351713048585785681933165, 7.35623950739432264766386568065, 8.249094780469195592028243824111
