| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.410403507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410403507\) |
| \(L(1)\) |
\(\approx\) |
\(1.382351709\) |
| \(L(1)\) |
\(\approx\) |
\(1.382351709\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 113 | \( 1 \) |
| 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 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.74407981079588034308742283993, −28.113235506810464825014406599253, −27.8295050492920567609617668079, −26.39305822669310418919189742968, −24.6787348329532344355989732958, −24.06269034605401740016192084110, −23.18982794827482109471165754337, −22.41097657182681279061086997101, −21.37471719936435248618500918751, −20.32429811472435491198537018112, −19.13578443626829025880364974845, −17.72658456343687436581920464978, −16.59482423032170609404657691121, −15.61549157335356787512434189318, −14.729318492424229331054931001565, −13.33566441793834352310003229381, −12.052737228135680196884800536063, −11.42928085710404959444406433149, −10.672844164914841214363882609218, −8.39306375972171956083202264783, −7.03739507972235900526552461764, −6.02525567517102938178010642713, −4.56153238121961788161172100373, −3.93473494741513833326731335557, −1.62199966077921185239494283423,
1.62199966077921185239494283423, 3.93473494741513833326731335557, 4.56153238121961788161172100373, 6.02525567517102938178010642713, 7.03739507972235900526552461764, 8.39306375972171956083202264783, 10.672844164914841214363882609218, 11.42928085710404959444406433149, 12.052737228135680196884800536063, 13.33566441793834352310003229381, 14.729318492424229331054931001565, 15.61549157335356787512434189318, 16.59482423032170609404657691121, 17.72658456343687436581920464978, 19.13578443626829025880364974845, 20.32429811472435491198537018112, 21.37471719936435248618500918751, 22.41097657182681279061086997101, 23.18982794827482109471165754337, 24.06269034605401740016192084110, 24.6787348329532344355989732958, 26.39305822669310418919189742968, 27.8295050492920567609617668079, 28.113235506810464825014406599253, 29.74407981079588034308742283993
