| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9589206551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9589206551\) |
| \(L(1)\) |
\(\approx\) |
\(0.7959530465\) |
| \(L(1)\) |
\(\approx\) |
\(0.7959530465\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 389 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.50522609423344634598301667043, −23.93303610951297120741112786019, −22.65137487236026023046660138115, −21.692770315622149562442732245763, −20.982730010483013084848069534538, −20.22826034557116208086662548319, −18.76719262303433522810550337267, −18.12607946190051524520556599949, −17.58246422693649903708331993715, −16.74503878350972065768681797807, −16.12769410524643100586589105404, −14.81174189138413842888695075140, −13.89687365495279379741255945, −12.49308034670086762398849129626, −11.57406387419414723337068291282, −10.948495168646352192001561299758, −9.9586012256740034860762082694, −9.21214411303056742194384568242, −8.00424695085173023958944451878, −6.944281492809192011261149890645, −5.961735988751133269942137719111, −5.315623561405507224705933835991, −3.65166942044686405038689236876, −1.7405639579637656931414491188, −1.26067806955053997073772743415,
1.26067806955053997073772743415, 1.7405639579637656931414491188, 3.65166942044686405038689236876, 5.315623561405507224705933835991, 5.961735988751133269942137719111, 6.944281492809192011261149890645, 8.00424695085173023958944451878, 9.21214411303056742194384568242, 9.9586012256740034860762082694, 10.948495168646352192001561299758, 11.57406387419414723337068291282, 12.49308034670086762398849129626, 13.89687365495279379741255945, 14.81174189138413842888695075140, 16.12769410524643100586589105404, 16.74503878350972065768681797807, 17.58246422693649903708331993715, 18.12607946190051524520556599949, 18.76719262303433522810550337267, 20.22826034557116208086662548319, 20.982730010483013084848069534538, 21.692770315622149562442732245763, 22.65137487236026023046660138115, 23.93303610951297120741112786019, 24.50522609423344634598301667043
