| L(s) = 1 | + i·2-s − 4-s − 4.70i·7-s − i·8-s − 4.70·11-s − i·13-s + 4.70·14-s + 16-s − 0.701i·17-s + 1.70·19-s − 4.70i·22-s + 26-s + 4.70i·28-s − 6.40·29-s − 10.1·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.77i·7-s − 0.353i·8-s − 1.41·11-s − 0.277i·13-s + 1.25·14-s + 0.250·16-s − 0.170i·17-s + 0.390·19-s − 1.00i·22-s + 0.196·26-s + 0.888i·28-s − 1.18·29-s − 1.81·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1663926201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1663926201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.70iT - 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 17 | \( 1 + 0.701iT - 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 1.70iT - 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 2.40iT - 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 6.40iT - 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.59iT - 97T^{2} \) |
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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
−8.079739905535036126416092181801, −7.46078717585583168456688796081, −7.32213042423795138395611291517, −6.39571229601425696406836764518, −5.37683820193734443211639999378, −5.05259643204581037110378371737, −3.92834629825321740830013087038, −3.54058451163298191438397200033, −2.25868507476286014158407198110, −0.909900953948142105451136225156,
0.04965785230790424375785183233, 1.75386455608993649491563654402, 2.38681239480376779599892567061, 3.06136439019000916426024322383, 3.97380703888279891457988498970, 5.17543946727932166329508234153, 5.38766298789659159898101835892, 6.11163086198830938584317163282, 7.24547519991211931414824921030, 7.992533745960045344410317665750
