L(s) = 1 | + i·3-s + (2.08 + 0.817i)5-s − 3.41i·7-s − 9-s + 5.18·11-s + 5.24i·13-s + (−0.817 + 2.08i)15-s − 7.93i·17-s + 1.59·19-s + 3.41·21-s − 7.16i·23-s + (3.66 + 3.40i)25-s − i·27-s + 6.58·29-s − 5.96·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.930 + 0.365i)5-s − 1.29i·7-s − 0.333·9-s + 1.56·11-s + 1.45i·13-s + (−0.211 + 0.537i)15-s − 1.92i·17-s + 0.366·19-s + 0.745·21-s − 1.49i·23-s + (0.732 + 0.680i)25-s − 0.192i·27-s + 1.22·29-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.518755626\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.518755626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.08 - 0.817i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 3.41iT - 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 13 | \( 1 - 5.24iT - 13T^{2} \) |
| 17 | \( 1 + 7.93iT - 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 + 7.16iT - 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 + 5.96T + 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 + 0.663iT - 43T^{2} \) |
| 47 | \( 1 + 7.08iT - 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 71 | \( 1 - 1.39T + 71T^{2} \) |
| 73 | \( 1 - 6.22iT - 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 4.62iT - 97T^{2} \) |
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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
−8.841169435813674535660120878114, −7.31691420792377514352596678890, −6.89691486590858108190036378572, −6.42579309766952533613948882155, −5.32468820391048675539681429558, −4.46273852854162946285812975762, −3.97689794254322665377606285285, −2.96319871169364041202832461111, −1.89722043444732609396296121935, −0.77531793749300556370000558209,
1.28457083334471427783624836864, 1.76903756596888634749497862436, 2.92447638493268212094984791043, 3.70733856996634512691974158178, 5.07178939822093742413695075149, 5.63342698728113647184777545737, 6.22656663412857461272197194958, 6.75414052367638760394817235908, 8.061386011959020146103357331019, 8.412780438064087503595071513133