L(s) = 1 | + (0.551 + 0.955i)5-s + (−1.62 + 2.81i)7-s + (1.28 − 2.23i)11-s + (−1.58 − 2.74i)13-s − 4.71·17-s + 5.75·19-s + (−2.35 − 4.07i)23-s + (1.89 − 3.27i)25-s + (−3.66 + 6.34i)29-s + (−2.93 − 5.07i)31-s − 3.58·35-s + 0.0714·37-s + (1.63 + 2.83i)41-s + (2.12 − 3.67i)43-s + (4.72 − 8.18i)47-s + ⋯ |
L(s) = 1 | + (0.246 + 0.427i)5-s + (−0.614 + 1.06i)7-s + (0.388 − 0.672i)11-s + (−0.440 − 0.762i)13-s − 1.14·17-s + 1.32·19-s + (−0.490 − 0.850i)23-s + (0.378 − 0.655i)25-s + (−0.680 + 1.17i)29-s + (−0.526 − 0.911i)31-s − 0.605·35-s + 0.0117·37-s + (0.255 + 0.443i)41-s + (0.323 − 0.560i)43-s + (0.689 − 1.19i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.182992286\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182992286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.551 - 0.955i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.62 - 2.81i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.28 + 2.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 + 2.74i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + (2.35 + 4.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.66 - 6.34i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.93 + 5.07i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.0714T + 37T^{2} \) |
| 41 | \( 1 + (-1.63 - 2.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.12 + 3.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.72 + 8.18i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + (4.19 + 7.26i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.66 - 8.07i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.09 + 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.335T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + (-4.85 + 8.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.07 - 5.31i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.42T + 89T^{2} \) |
| 97 | \( 1 + (-6.39 + 11.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.596351489088348930675779761806, −7.69434075631117808896579206207, −6.85669633846765710412306918609, −6.13050796165833040163718604014, −5.60790108420620910518591267221, −4.71273300712364325995850271089, −3.51204188346632907281369934794, −2.84348822376101739143782152385, −2.03742442853172833490960104054, −0.37873035128421365659439625722,
1.11434992447713438041942657945, 2.09296999414831741143325739495, 3.34271620245334976526231360988, 4.15622985596859531023408183531, 4.78316838245568022459279241830, 5.75490991096274939020642223006, 6.64003769251608665865238069279, 7.27631871641565367790427343832, 7.73941098542076794838114465502, 9.068975415175539397449325549630