| L(s) = 1 | + 2-s + 4-s − 1.41i·5-s + 2·7-s + 8-s − 1.41i·10-s − 1.41i·11-s + 2·14-s + 16-s − 1.41i·17-s + (1 + 4.24i)19-s − 1.41i·20-s − 1.41i·22-s − 1.41i·23-s + 2.99·25-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632i·5-s + 0.755·7-s + 0.353·8-s − 0.447i·10-s − 0.426i·11-s + 0.534·14-s + 0.250·16-s − 0.342i·17-s + (0.229 + 0.973i)19-s − 0.316i·20-s − 0.301i·22-s − 0.294i·23-s + 0.599·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07657 - 0.403678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07657 - 0.403678i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−11.60149405193355822763955140820, −10.80330306813993970833263470164, −9.659807264846467458778624943497, −8.496290412573662892174098455497, −7.73785105689983516372552747557, −6.45531484065039111759474564162, −5.34220918876125630643851219386, −4.57704547109430434719243518441, −3.26503421558058707756956709598, −1.56747283395371878099014640505,
1.97662533339914728359032500466, 3.34235297594197031787786287956, 4.59410887246697469962315618459, 5.55914704355960344271102917023, 6.80743545105583237261684220497, 7.51615291359582329351442121535, 8.713226341000717954830444417135, 9.933887292912997792071390664945, 10.97448967012839519454146936289, 11.45759246125891440600548733891
