L(s) = 1 | + (0.122 − 0.350i)2-s + (−0.733 + 0.680i)3-s + (0.673 + 0.537i)4-s + (0.148 + 0.340i)6-s + (0.586 − 0.368i)8-s + (0.0747 − 0.997i)9-s + (−0.859 + 0.0643i)12-s + (1.61 − 0.367i)13-s + (0.134 + 0.589i)16-s + (−0.340 − 0.148i)18-s + (−0.433 + 0.900i)23-s + (−0.179 + 0.668i)24-s + (0.623 − 0.781i)25-s + (0.0687 − 0.610i)26-s + (0.623 + 0.781i)27-s + ⋯ |
L(s) = 1 | + (0.122 − 0.350i)2-s + (−0.733 + 0.680i)3-s + (0.673 + 0.537i)4-s + (0.148 + 0.340i)6-s + (0.586 − 0.368i)8-s + (0.0747 − 0.997i)9-s + (−0.859 + 0.0643i)12-s + (1.61 − 0.367i)13-s + (0.134 + 0.589i)16-s + (−0.340 − 0.148i)18-s + (−0.433 + 0.900i)23-s + (−0.179 + 0.668i)24-s + (0.623 − 0.781i)25-s + (0.0687 − 0.610i)26-s + (0.623 + 0.781i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.268518289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268518289\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.733 - 0.680i)T \) |
| 23 | \( 1 + (0.433 - 0.900i)T \) |
| 29 | \( 1 + (-0.149 + 0.988i)T \) |
good | 2 | \( 1 + (-0.122 + 0.350i)T + (-0.781 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (-1.61 + 0.367i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 31 | \( 1 + (1.82 + 0.638i)T + (0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 41 | \( 1 + (-0.660 - 0.660i)T + iT^{2} \) |
| 43 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 47 | \( 1 + (-0.631 + 1.00i)T + (-0.433 - 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - 1.24iT - T^{2} \) |
| 61 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (-0.443 - 1.94i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.754 - 0.264i)T + (0.781 - 0.623i)T^{2} \) |
| 79 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (0.974 - 0.222i)T^{2} \) |
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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
−9.542646946136784128428528328572, −8.665764241666222553709583254404, −7.84541796305862054447849146429, −6.93443504168911231400811946041, −6.08261185634416217160549344032, −5.55753101300473537555538390525, −4.16795060323313123612581655997, −3.78740572587848711894349421088, −2.72840122731260673378784348655, −1.30561391527506769766683580055,
1.22837660014012224975507460414, 2.05480184107074485419237080698, 3.45479131631686313925908564750, 4.74387833026659074891553275011, 5.55428119908729858040754711410, 6.17988890134589581737059399784, 6.83421403194898408802116893695, 7.43395933431584256430793307641, 8.385846110472302715264756464034, 9.168999110812529625591603057567