| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.681272646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.681272646\) |
| \(L(1)\) |
\(\approx\) |
\(2.855239499\) |
| \(L(1)\) |
\(\approx\) |
\(2.855239499\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
| 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 | \( 1 - T \) |
| 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
−18.57461442709341223998967133665, −18.12867267955007496081691410066, −16.65285567440518928114386514807, −16.25610147994830901209726597634, −15.33670521593108582187643860710, −15.12490014997311251098790858583, −14.43034712197013933097609669088, −13.724099892511429493235984951224, −13.12810690429612653785251464531, −12.46475010014542513263326283844, −11.651106191131539592653562492169, −11.035886439707883637094400585366, −10.47454557221647823609131398065, −9.40323449567117887585132291395, −8.37274791795417712584987878569, −7.88640867288464697636139522933, −7.45712642343535876437049664941, −6.66611406697288217421303426204, −5.2729618688142222906983841244, −5.08253018871779565846168282744, −4.018205732080327339231579514898, −3.40961645693790757429275014625, −2.95312144487582789266286814231, −1.80929254399465482209805615269, −1.16558507272148551350484373853,
1.16558507272148551350484373853, 1.80929254399465482209805615269, 2.95312144487582789266286814231, 3.40961645693790757429275014625, 4.018205732080327339231579514898, 5.08253018871779565846168282744, 5.2729618688142222906983841244, 6.66611406697288217421303426204, 7.45712642343535876437049664941, 7.88640867288464697636139522933, 8.37274791795417712584987878569, 9.40323449567117887585132291395, 10.47454557221647823609131398065, 11.035886439707883637094400585366, 11.651106191131539592653562492169, 12.46475010014542513263326283844, 13.12810690429612653785251464531, 13.724099892511429493235984951224, 14.43034712197013933097609669088, 15.12490014997311251098790858583, 15.33670521593108582187643860710, 16.25610147994830901209726597634, 16.65285567440518928114386514807, 18.12867267955007496081691410066, 18.57461442709341223998967133665
