| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1747 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1747 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4204392599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4204392599\) |
| \(L(1)\) |
\(\approx\) |
\(0.3758144066\) |
| \(L(1)\) |
\(\approx\) |
\(0.3758144066\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1747 | \( 1 \) |
| 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
−19.782920749073877579656423370313, −19.03840054269251259480147200044, −18.83588249333759973761065787981, −17.814427429800302969892776132916, −17.143013792293349165288421189398, −16.37667914242129259754031037040, −15.810634257763446946357248365626, −15.50316265596569064921013077663, −14.33535414268214585636177310258, −12.85127767584122743157340209333, −12.405766825521496299006106114276, −11.74969604415455247505583875617, −10.9968869843666726492548942143, −10.10902380592643131533733447432, −9.81566510449068839425347092384, −8.66522931176861878586905926816, −7.53155689602168396320373482417, −7.34619485696713996868092086132, −6.40344962363615059887279358756, −5.47510055897699039062223266345, −4.6624684367315182370446833772, −3.29296880294116849376123946213, −2.72944469513069855647428099637, −1.11431567729611955103296918545, −0.39748870359136916325317683135,
0.39748870359136916325317683135, 1.11431567729611955103296918545, 2.72944469513069855647428099637, 3.29296880294116849376123946213, 4.6624684367315182370446833772, 5.47510055897699039062223266345, 6.40344962363615059887279358756, 7.34619485696713996868092086132, 7.53155689602168396320373482417, 8.66522931176861878586905926816, 9.81566510449068839425347092384, 10.10902380592643131533733447432, 10.9968869843666726492548942143, 11.74969604415455247505583875617, 12.405766825521496299006106114276, 12.85127767584122743157340209333, 14.33535414268214585636177310258, 15.50316265596569064921013077663, 15.810634257763446946357248365626, 16.37667914242129259754031037040, 17.143013792293349165288421189398, 17.814427429800302969892776132916, 18.83588249333759973761065787981, 19.03840054269251259480147200044, 19.782920749073877579656423370313
