| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s − 2.44i·6-s + (6.69 − 2.03i)7-s − 2.82i·8-s + 2.99·9-s − 5.30·11-s + 3.46·12-s − 15.0·13-s + (2.87 + 9.47i)14-s + 4.00·16-s − 1.80·17-s + 4.24i·18-s − 7.35i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.956 − 0.290i)7-s − 0.353i·8-s + 0.333·9-s − 0.482·11-s + 0.288·12-s − 1.15·13-s + (0.205 + 0.676i)14-s + 0.250·16-s − 0.106·17-s + 0.235i·18-s − 0.387i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.020457631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.020457631\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.69 + 2.03i)T \) |
| good | 11 | \( 1 + 5.30T + 121T^{2} \) |
| 13 | \( 1 + 15.0T + 169T^{2} \) |
| 17 | \( 1 + 1.80T + 289T^{2} \) |
| 19 | \( 1 + 7.35iT - 361T^{2} \) |
| 23 | \( 1 - 15.8iT - 529T^{2} \) |
| 29 | \( 1 - 44.3T + 841T^{2} \) |
| 31 | \( 1 - 6.10iT - 961T^{2} \) |
| 37 | \( 1 - 36.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 26.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 38.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 90.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 34.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 58.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 140.T + 9.40e3T^{2} \) |
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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
−10.10068078041708921310088108348, −9.113919272534655678919481787672, −8.145647942865728020889733577216, −7.46943619591177407525496805907, −6.76807838582271805461496960727, −5.67063557184129759980131081049, −4.91134037693667120707964404193, −4.31985640606888476346915143651, −2.69333210667230283251663786146, −1.12882008801445061657184531953,
0.38713532594252909456652273361, 1.83500491848749460357307164765, 2.78070601790374389263775563229, 4.27476370983115842853312463103, 4.94424550853723820657693402025, 5.71268659507969885410062420658, 6.92894622286338738622980100112, 7.88494365940727107222732727064, 8.609200820322838172545725199966, 9.647444259561768912074398489942
