L(s) = 1 | + 1.87·2-s − 1.28·3-s + 2.53·4-s − 0.684·5-s − 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s − 1.28·10-s − 11-s − 3.25·12-s + 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s + 1.96·17-s + 1.22·18-s − 1.73·20-s + 1.28·21-s − 1.87·22-s + 23-s − 3.70·24-s − 0.532·25-s + 3.25·26-s + 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯ |
L(s) = 1 | + 1.87·2-s − 1.28·3-s + 2.53·4-s − 0.684·5-s − 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s − 1.28·10-s − 11-s − 3.25·12-s + 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s + 1.96·17-s + 1.22·18-s − 1.73·20-s + 1.28·21-s − 1.87·22-s + 23-s − 3.70·24-s − 0.532·25-s + 3.25·26-s + 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.309779593\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.309779593\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 - 1.87T + T^{2} \) |
| 3 | \( 1 + 1.28T + T^{2} \) |
| 5 | \( 1 + 0.684T + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 - 1.96T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.96T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.28T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.559604368324480836732841735456, −7.64618333855100589292223577107, −6.89097296964873019192354110756, −6.18811093351281029212212119672, −5.70639191809599558408664393814, −5.15284873771072311282724014825, −4.24918905050462287344085091642, −3.36696554176594540161337435032, −2.96619327250389796491487991684, −1.15314755041358650459711425266,
1.15314755041358650459711425266, 2.96619327250389796491487991684, 3.36696554176594540161337435032, 4.24918905050462287344085091642, 5.15284873771072311282724014825, 5.70639191809599558408664393814, 6.18811093351281029212212119672, 6.89097296964873019192354110756, 7.64618333855100589292223577107, 8.559604368324480836732841735456