L(s) = 1 | + (1.80 + 3.12i)5-s + (1.02 + 2.44i)7-s + (3.01 − 5.21i)11-s + (−2.55 + 4.42i)13-s + (−0.111 − 0.192i)17-s + (−1.71 + 2.96i)19-s + (0.509 + 0.883i)23-s + (−3.99 + 6.91i)25-s + (2.83 + 4.91i)29-s + 5.04·31-s + (−5.77 + 7.58i)35-s + (1.68 − 2.91i)37-s + (−0.0955 + 0.165i)41-s + (−1.71 − 2.96i)43-s − 2.07·47-s + ⋯ |
L(s) = 1 | + (0.805 + 1.39i)5-s + (0.386 + 0.922i)7-s + (0.908 − 1.57i)11-s + (−0.709 + 1.22i)13-s + (−0.0269 − 0.0466i)17-s + (−0.392 + 0.680i)19-s + (0.106 + 0.184i)23-s + (−0.798 + 1.38i)25-s + (0.526 + 0.912i)29-s + 0.906·31-s + (−0.976 + 1.28i)35-s + (0.277 − 0.479i)37-s + (−0.0149 + 0.0258i)41-s + (−0.261 − 0.452i)43-s − 0.301·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.169209753\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.169209753\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.02 - 2.44i)T \) |
good | 5 | \( 1 + (-1.80 - 3.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.01 + 5.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 - 4.42i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.111 + 0.192i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.71 - 2.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.509 - 0.883i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.83 - 4.91i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.04T + 31T^{2} \) |
| 37 | \( 1 + (-1.68 + 2.91i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0955 - 0.165i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.71 + 2.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.07T + 47T^{2} \) |
| 53 | \( 1 + (-2.65 - 4.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 7.59T + 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 + (-6.30 - 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + (3.59 + 6.23i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.44 + 7.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.72 + 4.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.929113566929956401971474244069, −8.396913504692167803804643631054, −7.25131688056337848960390551752, −6.53582565978545072194151987296, −6.07999350468142701203693939725, −5.34861492271645541497221341033, −4.16048218014906746563448025998, −3.16143265743516086627334890832, −2.45674824848508136304517200305, −1.52564500754542976905003842655,
0.68743174603250971774640484618, 1.57467895205341078722583745391, 2.56062626406603579218045832358, 4.07764339671576234313590417345, 4.71647284489462342577992845498, 5.11700113987402742435632927575, 6.25327437416310454183341793986, 6.99292281432361969386153810883, 7.84689625602800398437322258295, 8.466379543567900421130805133899