| L(s) = 1 | + i·2-s + 0.987·3-s − 4-s + 0.987i·6-s + 4.78·7-s − i·8-s − 2.02·9-s − 5.98·11-s − 0.987·12-s − 3.49i·13-s + 4.78i·14-s + 16-s − 4.96i·17-s − 2.02i·18-s − 7.33i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.570·3-s − 0.5·4-s + 0.403i·6-s + 1.81·7-s − 0.353i·8-s − 0.674·9-s − 1.80·11-s − 0.285·12-s − 0.969i·13-s + 1.28i·14-s + 0.250·16-s − 1.20i·17-s − 0.477i·18-s − 1.68i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.563013969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.563013969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + (4.61 - 3.96i)T \) |
| good | 3 | \( 1 - 0.987T + 3T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 + 3.49iT - 13T^{2} \) |
| 17 | \( 1 + 4.96iT - 17T^{2} \) |
| 19 | \( 1 + 7.33iT - 19T^{2} \) |
| 23 | \( 1 + 1.74iT - 23T^{2} \) |
| 29 | \( 1 + 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 41 | \( 1 + 0.530T + 41T^{2} \) |
| 43 | \( 1 - 1.76iT - 43T^{2} \) |
| 47 | \( 1 - 4.30T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 - 2.15iT - 59T^{2} \) |
| 61 | \( 1 + 3.06iT - 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 - 9.05T + 73T^{2} \) |
| 79 | \( 1 - 5.56iT - 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 - 8.45iT - 89T^{2} \) |
| 97 | \( 1 + 3.64iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.756507853131345003878207635434, −8.194235946617013654301705351757, −7.79000608843266490803384586015, −7.07009451934944312191583753175, −5.64948309218779137183290702876, −5.10774528455045497398586722671, −4.58862775423316071915925857518, −2.96961015813447315188362498389, −2.35575186462187126066753702032, −0.50645560554474018228869057630,
1.66898257258077930958981526217, 2.16535700953362817604848466571, 3.38617380065546857511518537084, 4.28485873422194419708039321559, 5.25536054172961695494373557007, 5.76543210130486816136516659175, 7.41117511645337644644882098514, 8.110178071330921896375648117352, 8.403968009294579600891193760605, 9.239417615861162616411613296429
