| L(s) = 1 | + 1.81i·3-s + i·5-s − 2.60·7-s − 0.312·9-s − 1.78·11-s + 1.16·13-s − 1.81·15-s + 5.52i·17-s − 1.77·19-s − 4.74i·21-s + (−3.56 − 3.20i)23-s − 25-s + 4.89i·27-s − 3.00·29-s − 9.26i·31-s + ⋯ |
| L(s) = 1 | + 1.05i·3-s + 0.447i·5-s − 0.986·7-s − 0.104·9-s − 0.536·11-s + 0.321·13-s − 0.469·15-s + 1.34i·17-s − 0.407·19-s − 1.03i·21-s + (−0.743 − 0.668i)23-s − 0.200·25-s + 0.941i·27-s − 0.558·29-s − 1.66i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09894708324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09894708324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (3.56 + 3.20i)T \) |
| good | 3 | \( 1 - 1.81iT - 3T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 - 5.52iT - 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 31 | \( 1 + 9.26iT - 31T^{2} \) |
| 37 | \( 1 - 5.45iT - 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 - 0.268iT - 47T^{2} \) |
| 53 | \( 1 + 3.86iT - 53T^{2} \) |
| 59 | \( 1 + 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 6.14iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 4.90iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 0.0944T + 79T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 + 3.54iT - 89T^{2} \) |
| 97 | \( 1 + 8.13iT - 97T^{2} \) |
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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
−8.383327471154607994342106842417, −7.74613364976974269589401018475, −6.64043684778594972425787965517, −6.17906463615000720475549894059, −5.34746058412181281809820076490, −4.26868984246747753598029047056, −3.80387634566506784292866367022, −2.99165627456848266338843608536, −1.91484750479182882252772902433, −0.03065800547838597859615353166,
1.10418473238930235080746761820, 2.16911681198587284260237017419, 3.07516164882023988094477785180, 4.02902067878658714349550491448, 5.08851040390035893831340119311, 5.82958285388320305767891331845, 6.63602674013548180803655981296, 7.19914838336828838336206982992, 7.82758595788830275091139600606, 8.641565860151999051671802130483
