| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7399171818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7399171818\) |
| \(L(1)\) |
\(\approx\) |
\(0.6317596835\) |
| \(L(1)\) |
\(\approx\) |
\(0.6317596835\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2003 | \( 1 \) |
| 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
−19.80677488058368093848112545306, −19.00179763923805526032866374122, −18.444894109303576025115655063576, −18.02233939221764393633652218528, −16.48571713028721824339451019305, −16.15790809458729041345731379013, −15.40668386005940036352821013012, −15.19224505365749091568735154851, −13.81468560838389163340501332734, −13.15840925279941569193390419390, −12.36311016671627391955797469930, −11.510494059436633749811744641674, −10.59795633407780787621306805007, −10.05900924507203204141580677388, −9.02422402137435810626380613532, −8.67897308820909355379483783733, −7.76706122569506530675629788892, −7.26789055150186317574822898631, −6.49659579172612239862542396876, −5.36628670054273651200785500510, −3.880605391603492829764035566797, −3.42397834535261368054495684074, −2.59052907614924315373160750849, −1.62863397378040708101418642961, −0.37490716172147350925175825523,
0.37490716172147350925175825523, 1.62863397378040708101418642961, 2.59052907614924315373160750849, 3.42397834535261368054495684074, 3.880605391603492829764035566797, 5.36628670054273651200785500510, 6.49659579172612239862542396876, 7.26789055150186317574822898631, 7.76706122569506530675629788892, 8.67897308820909355379483783733, 9.02422402137435810626380613532, 10.05900924507203204141580677388, 10.59795633407780787621306805007, 11.510494059436633749811744641674, 12.36311016671627391955797469930, 13.15840925279941569193390419390, 13.81468560838389163340501332734, 15.19224505365749091568735154851, 15.40668386005940036352821013012, 16.15790809458729041345731379013, 16.48571713028721824339451019305, 18.02233939221764393633652218528, 18.444894109303576025115655063576, 19.00179763923805526032866374122, 19.80677488058368093848112545306
