| L(s) = 1 | + (0.0607 + 1.99i)2-s + (−3.99 + 0.242i)4-s + 5.82·5-s + 5.45i·7-s + (−0.728 − 7.96i)8-s + (0.353 + 11.6i)10-s − 1.60i·11-s + 23.5·13-s + (−10.8 + 0.331i)14-s + (15.8 − 1.94i)16-s + 5.92·17-s − 4.35i·19-s + (−23.2 + 1.41i)20-s + (3.21 − 0.0977i)22-s − 26.6i·23-s + ⋯ |
| L(s) = 1 | + (0.0303 + 0.999i)2-s + (−0.998 + 0.0607i)4-s + 1.16·5-s + 0.778i·7-s + (−0.0910 − 0.995i)8-s + (0.0353 + 1.16i)10-s − 0.146i·11-s + 1.80·13-s + (−0.778 + 0.0236i)14-s + (0.992 − 0.121i)16-s + 0.348·17-s − 0.229i·19-s + (−1.16 + 0.0707i)20-s + (0.146 − 0.00444i)22-s − 1.15i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0607 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.240798653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240798653\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0607 - 1.99i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 5.82T + 25T^{2} \) |
| 7 | \( 1 - 5.45iT - 49T^{2} \) |
| 11 | \( 1 + 1.60iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 - 5.92T + 289T^{2} \) |
| 23 | \( 1 + 26.6iT - 529T^{2} \) |
| 29 | \( 1 - 1.49T + 841T^{2} \) |
| 31 | \( 1 - 31.3iT - 961T^{2} \) |
| 37 | \( 1 - 26.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 76.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 49.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 116.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 65.7T + 9.40e3T^{2} \) |
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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
−10.28410921884212142181342485920, −9.346362632947545059894153345250, −8.763549017774886762851914853208, −8.025901458854075315181495187410, −6.64686956832354574008365203255, −6.04268017798177852042612932198, −5.46446475897937478957410845766, −4.24959241281297372535626666189, −2.86905371698358830308076208245, −1.24016586777330062866961572871,
1.00987948362913085741370143414, 1.95433069607071373033709114464, 3.37211313278926520750592978049, 4.21006838867579816199233019868, 5.54641576611563379182663785662, 6.16472540902529320855069790716, 7.59941413262134987284164242337, 8.596955422888902838409669164765, 9.523437027569596076340446054304, 10.03852995932606820651870799239
