| L(s) = 1 | − i·2-s − 3.30i·3-s − 4-s − 3.30·6-s − 2.60i·7-s + i·8-s − 7.90·9-s − 2.30·11-s + 3.30i·12-s − 1.30i·13-s − 2.60·14-s + 16-s − 6i·17-s + 7.90i·18-s − 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.90i·3-s − 0.5·4-s − 1.34·6-s − 0.984i·7-s + 0.353i·8-s − 2.63·9-s − 0.694·11-s + 0.953i·12-s − 0.361i·13-s − 0.696·14-s + 0.250·16-s − 1.45i·17-s + 1.86i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8836342355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8836342355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 3 | \( 1 + 3.30iT - 3T^{2} \) |
| 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 1.30iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.90iT - 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 + 0.605iT - 43T^{2} \) |
| 47 | \( 1 - 4.60iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.51iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78iT - 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4iT - 97T^{2} \) |
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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
−8.401886326933767201963264357225, −7.72135383188643366673300531933, −7.18532353862061470008727743972, −6.37916493416222010322830762330, −5.43602758158377087054906794543, −4.40597362503277572973340579017, −2.95965143476785906450790399701, −2.43925672926308053463527222283, −1.14046419582047499651588051164, −0.35897365762615882054820879967,
2.39386884788498814711500129560, 3.50602512263894727306296580840, 4.26247640333869065064156096916, 5.09717352272085521908028714941, 5.71868154862360284281038166984, 6.35398749140177310297830914921, 7.84454553961700542486236218614, 8.527273918355052946531069953506, 9.067297388263805762324121347360, 9.750713744738748517869496790594
