| L(s) = 1 | + i·2-s + 1.77i·3-s − 4-s − 1.77·6-s − 2.69i·7-s − i·8-s − 0.144·9-s + 5.54·11-s − 1.77i·12-s + 2.91i·13-s + 2.69·14-s + 16-s + 4.91i·17-s − 0.144i·18-s − 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.02i·3-s − 0.5·4-s − 0.723·6-s − 1.01i·7-s − 0.353i·8-s − 0.0483·9-s + 1.67·11-s − 0.511i·12-s + 0.809i·13-s + 0.719·14-s + 0.250·16-s + 1.19i·17-s − 0.0341i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871625 + 1.41032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871625 + 1.41032i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.77iT - 3T^{2} \) |
| 7 | \( 1 + 2.69iT - 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 2.91iT - 13T^{2} \) |
| 17 | \( 1 - 4.91iT - 17T^{2} \) |
| 23 | \( 1 + 3.60iT - 23T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 4.54iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.54iT - 43T^{2} \) |
| 47 | \( 1 + 0.836iT - 47T^{2} \) |
| 53 | \( 1 + 9.78iT - 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 2.85iT - 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 5.15iT - 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.71iT - 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−10.14478038990527302647938908680, −9.468598072593009167919550115519, −8.752874293113246485128328649730, −7.81269380801745631115106418347, −6.61413045910446523553128376259, −6.36635678655076905848975237776, −4.73437425656294287503114842348, −4.23274293104942551334939936796, −3.57170476355457033795059513807, −1.37384753281162657188321274580,
0.936750481410795245148169208529, 2.03683891189127788840186396488, 3.06556509149052843203318753050, 4.28615626997913101189517988385, 5.49294426597085922542007122156, 6.35748611916909983362952119576, 7.24851569276400909066275386894, 8.158017599197913552061478965094, 9.117880400138790770687371730668, 9.568467128120567785415241232297
