| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.661503236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661503236\) |
| \(L(1)\) |
\(\approx\) |
\(1.775555086\) |
| \(L(1)\) |
\(\approx\) |
\(1.775555086\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1049 | \( 1 \) |
| 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
−21.88566585708318691710740261205, −21.01839549653672684483888289626, −20.16258916864251108405817621929, −19.34300595938663735430442080531, −18.2080735460654961813314736950, −17.57926111624711643436766359858, −16.57571292869275245509345756747, −16.14215627049168521183073020526, −15.40068890331067409472331934748, −14.159754902989861834015493239495, −13.54805025295891226543979229006, −12.90925231635459848673677532890, −12.08418937113831456391929151933, −11.37473738808144694406820336944, −10.44995067275579710191254782704, −9.817939033559673475515954311014, −8.783597471220108387033210607264, −7.0919038575914814137606934245, −6.54638221726515344843041656756, −5.97802514969606882941933821753, −5.26622997641575764832960121987, −4.15594779318312896027878890396, −3.38893856313143913010188465890, −2.0767157481203730292469415706, −1.147421865867570846455807381290,
1.147421865867570846455807381290, 2.0767157481203730292469415706, 3.38893856313143913010188465890, 4.15594779318312896027878890396, 5.26622997641575764832960121987, 5.97802514969606882941933821753, 6.54638221726515344843041656756, 7.0919038575914814137606934245, 8.783597471220108387033210607264, 9.817939033559673475515954311014, 10.44995067275579710191254782704, 11.37473738808144694406820336944, 12.08418937113831456391929151933, 12.90925231635459848673677532890, 13.54805025295891226543979229006, 14.159754902989861834015493239495, 15.40068890331067409472331934748, 16.14215627049168521183073020526, 16.57571292869275245509345756747, 17.57926111624711643436766359858, 18.2080735460654961813314736950, 19.34300595938663735430442080531, 20.16258916864251108405817621929, 21.01839549653672684483888289626, 21.88566585708318691710740261205
