| L(s) = 1 | + 2-s − 2.29·3-s + 4-s − 2.85·5-s − 2.29·6-s + 3.49·7-s + 8-s + 2.27·9-s − 2.85·10-s − 4.96·11-s − 2.29·12-s − 4.70·13-s + 3.49·14-s + 6.56·15-s + 16-s + 3.17·17-s + 2.27·18-s − 19-s − 2.85·20-s − 8.03·21-s − 4.96·22-s + 3.65·23-s − 2.29·24-s + 3.15·25-s − 4.70·26-s + 1.66·27-s + 3.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s − 1.27·5-s − 0.937·6-s + 1.32·7-s + 0.353·8-s + 0.758·9-s − 0.903·10-s − 1.49·11-s − 0.662·12-s − 1.30·13-s + 0.934·14-s + 1.69·15-s + 0.250·16-s + 0.769·17-s + 0.536·18-s − 0.229·19-s − 0.638·20-s − 1.75·21-s − 1.05·22-s + 0.762·23-s − 0.468·24-s + 0.631·25-s − 0.923·26-s + 0.320·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 + 8.13T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 + 9.42T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 - 5.50T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 8.13T + 97T^{2} \) |
| 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
−7.42437668472874852875518625184, −6.83646203313965569791425214841, −5.83259090079933661481140568347, −5.14418884549092056680865399609, −4.81969863141205343814409884111, −4.35198672490220165025437692910, −3.15612654785905276933184851597, −2.38926481826028794568513627331, −1.04208983817882778912533675673, 0,
1.04208983817882778912533675673, 2.38926481826028794568513627331, 3.15612654785905276933184851597, 4.35198672490220165025437692910, 4.81969863141205343814409884111, 5.14418884549092056680865399609, 5.83259090079933661481140568347, 6.83646203313965569791425214841, 7.42437668472874852875518625184
