| L(s) = 1 | − 2.10·3-s + 1.44·9-s + 1.77·11-s − 1.44·13-s − 5.86·17-s + 7.16·19-s − 0.663·23-s + 3.27·27-s + 6.45·29-s − 10.0·31-s − 3.74·33-s − 5.01·37-s + 3.04·39-s − 1.92·41-s + 5.81·43-s − 3.70·47-s + 12.3·51-s − 0.592·53-s − 15.0·57-s − 7.59·59-s + 8.72·61-s − 2.16·67-s + 1.39·69-s + 6.49·71-s + 15.5·73-s − 2.70·79-s − 11.2·81-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 0.481·9-s + 0.535·11-s − 0.400·13-s − 1.42·17-s + 1.64·19-s − 0.138·23-s + 0.631·27-s + 1.19·29-s − 1.80·31-s − 0.651·33-s − 0.824·37-s + 0.487·39-s − 0.300·41-s + 0.887·43-s − 0.540·47-s + 1.73·51-s − 0.0813·53-s − 1.99·57-s − 0.988·59-s + 1.11·61-s − 0.264·67-s + 0.168·69-s + 0.771·71-s + 1.81·73-s − 0.304·79-s − 1.24·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.10T + 3T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 7.16T + 19T^{2} \) |
| 23 | \( 1 + 0.663T + 23T^{2} \) |
| 29 | \( 1 - 6.45T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.592T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 + 2.16T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 6.35T + 83T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
| 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
−7.01722422260066210694019549318, −6.72522888574893262130519540524, −5.91788686144529647827350293115, −5.27457421983362284700406420512, −4.78718915605117365831334860881, −3.94174044395445024558148622075, −3.06757125698205839305019399702, −2.03455328652303732293894007072, −1.00971610856382866765351562421, 0,
1.00971610856382866765351562421, 2.03455328652303732293894007072, 3.06757125698205839305019399702, 3.94174044395445024558148622075, 4.78718915605117365831334860881, 5.27457421983362284700406420512, 5.91788686144529647827350293115, 6.72522888574893262130519540524, 7.01722422260066210694019549318
