L(s) = 1 | − 2-s + 4-s + 5-s − 2.59·7-s − 8-s − 10-s − 4.74·11-s + 6.69·13-s + 2.59·14-s + 16-s + 0.748·17-s − 3.34·19-s + 20-s + 4.74·22-s − 1.49·23-s + 25-s − 6.69·26-s − 2.59·28-s − 3.94·29-s + 7.79·31-s − 32-s − 0.748·34-s − 2.59·35-s − 37-s + 3.34·38-s − 40-s + 6.44·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.981·7-s − 0.353·8-s − 0.316·10-s − 1.43·11-s + 1.85·13-s + 0.694·14-s + 0.250·16-s + 0.181·17-s − 0.767·19-s + 0.223·20-s + 1.01·22-s − 0.312·23-s + 0.200·25-s − 1.31·26-s − 0.490·28-s − 0.732·29-s + 1.39·31-s − 0.176·32-s − 0.128·34-s − 0.439·35-s − 0.164·37-s + 0.542·38-s − 0.158·40-s + 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.332331916190200413365636367693, −7.70007876145339272672595547563, −6.66622692625480793092338335561, −6.11581992695415511204386958984, −5.53938086533927244150498816698, −4.27466615243464645162382465458, −3.24704017252383920070718024178, −2.53151508336947485107452488907, −1.34827461296543348290549238663, 0,
1.34827461296543348290549238663, 2.53151508336947485107452488907, 3.24704017252383920070718024178, 4.27466615243464645162382465458, 5.53938086533927244150498816698, 6.11581992695415511204386958984, 6.66622692625480793092338335561, 7.70007876145339272672595547563, 8.332331916190200413365636367693