L(s) = 1 | − 0.268·2-s + 2.35·3-s − 1.92·4-s + 2.80·5-s − 0.631·6-s − 2.38·7-s + 1.05·8-s + 2.52·9-s − 0.753·10-s + 11-s − 4.53·12-s + 0.640·14-s + 6.60·15-s + 3.57·16-s − 1.83·17-s − 0.678·18-s + 8.01·19-s − 5.41·20-s − 5.60·21-s − 0.268·22-s + 5.66·23-s + 2.47·24-s + 2.88·25-s − 1.10·27-s + 4.59·28-s − 4.70·29-s − 1.77·30-s + ⋯ |
L(s) = 1 | − 0.189·2-s + 1.35·3-s − 0.963·4-s + 1.25·5-s − 0.257·6-s − 0.901·7-s + 0.372·8-s + 0.842·9-s − 0.238·10-s + 0.301·11-s − 1.30·12-s + 0.171·14-s + 1.70·15-s + 0.893·16-s − 0.445·17-s − 0.159·18-s + 1.83·19-s − 1.21·20-s − 1.22·21-s − 0.0572·22-s + 1.18·23-s + 0.505·24-s + 0.576·25-s − 0.213·27-s + 0.868·28-s − 0.873·29-s − 0.323·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.468065259\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.468065259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.268T + 2T^{2} \) |
| 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 8.01T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 9.28T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 + 7.92T + 61T^{2} \) |
| 67 | \( 1 - 0.663T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 2.81T + 73T^{2} \) |
| 79 | \( 1 - 9.63T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 3.07T + 89T^{2} \) |
| 97 | \( 1 - 1.53T + 97T^{2} \) |
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
−9.271817145287773340573607322847, −8.864074733877246528186151861630, −7.77856141811908098490193448011, −7.09700331941512478147868412371, −5.92933922016506512741403331729, −5.26435857832639313366251640357, −4.03835169220121623015155173401, −3.22745875259817960855736685905, −2.40386701631716552711210165179, −1.10959997272579234826762890770,
1.10959997272579234826762890770, 2.40386701631716552711210165179, 3.22745875259817960855736685905, 4.03835169220121623015155173401, 5.26435857832639313366251640357, 5.92933922016506512741403331729, 7.09700331941512478147868412371, 7.77856141811908098490193448011, 8.864074733877246528186151861630, 9.271817145287773340573607322847