L(s) = 1 | + 6.22i·2-s − 6.78·4-s − 105.·5-s + 86.6i·7-s + 157. i·8-s − 659. i·10-s − 110.·11-s − 288.·13-s − 539.·14-s − 1.19e3·16-s − 432.·17-s − 503. i·19-s + 718.·20-s − 687. i·22-s + (−2.08e3 + 1.45e3i)23-s + ⋯ |
L(s) = 1 | + 1.10i·2-s − 0.211·4-s − 1.89·5-s + 0.668i·7-s + 0.867i·8-s − 2.08i·10-s − 0.274·11-s − 0.473·13-s − 0.735·14-s − 1.16·16-s − 0.362·17-s − 0.319i·19-s + 0.401·20-s − 0.302i·22-s + (−0.820 + 0.571i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3841771626\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3841771626\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (2.08e3 - 1.45e3i)T \) |
good | 2 | \( 1 - 6.22iT - 32T^{2} \) |
| 5 | \( 1 + 105.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 86.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 110.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 288.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 432.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 503. iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 3.39e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.31e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.42e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.00e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.37e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.99e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.91e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 4.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.28e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.27e5iT - 8.58e9T^{2} \) |
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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
−11.79394072896009299598954810887, −10.66433741013977705124443144112, −9.001131533209039666712565402340, −8.079734845166537913131702004049, −7.54162230574368439990659467752, −6.51252189203315054522980554902, −5.21474149657918436281985530617, −4.12522656677455091975162844674, −2.59772270929847720504700064134, −0.14615753735100633175821095831,
0.941695585492235441746986803185, 2.74978590799708117554626744306, 3.83652372919257408167295597707, 4.56590633294736123166204793108, 6.73420577680759619248432164773, 7.59203106464740683278379413479, 8.545179799474574945844661377286, 10.05568035838428356325475893020, 10.77535246379865205988353159819, 11.63573089099781077722528984627