L(s) = 1 | + (0.186 − 0.215i)3-s + (0.723 + 0.690i)4-s + (−1.67 + 1.07i)5-s + (0.130 + 0.909i)9-s + (−0.888 − 0.458i)11-s + (0.283 − 0.0270i)12-s + (−0.0806 + 0.560i)15-s + (0.0475 + 0.998i)16-s + (−1.95 − 0.376i)20-s + (0.428 + 1.23i)23-s + (1.23 − 2.69i)25-s + (0.459 + 0.295i)27-s + (−0.911 + 1.28i)31-s + (−0.264 + 0.105i)33-s + (−0.533 + 0.748i)36-s + (0.786 − 1.36i)37-s + ⋯ |
L(s) = 1 | + (0.186 − 0.215i)3-s + (0.723 + 0.690i)4-s + (−1.67 + 1.07i)5-s + (0.130 + 0.909i)9-s + (−0.888 − 0.458i)11-s + (0.283 − 0.0270i)12-s + (−0.0806 + 0.560i)15-s + (0.0475 + 0.998i)16-s + (−1.95 − 0.376i)20-s + (0.428 + 1.23i)23-s + (1.23 − 2.69i)25-s + (0.459 + 0.295i)27-s + (−0.911 + 1.28i)31-s + (−0.264 + 0.105i)33-s + (−0.533 + 0.748i)36-s + (0.786 − 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8226225539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8226225539\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.888 + 0.458i)T \) |
| 67 | \( 1 + (0.327 - 0.945i)T \) |
good | 2 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 3 | \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \) |
| 5 | \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 13 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 17 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 19 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 23 | \( 1 + (-0.428 - 1.23i)T + (-0.786 + 0.618i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \) |
| 37 | \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + (-1.82 - 0.351i)T + (0.928 + 0.371i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 71 | \( 1 + (-0.601 - 0.573i)T + (0.0475 + 0.998i)T^{2} \) |
| 73 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 79 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 83 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 89 | \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.95073667370760698708871517328, −10.42415327798158492035987597726, −8.686151936316664394434540220497, −7.944107105061357402522520050884, −7.37641508520405151592267500679, −6.95336874162656302337800100761, −5.48562487770839808952568820819, −4.04530290148030246159685521429, −3.23748138422964453597294768692, −2.39538959573792573528500386974,
0.873462383892445419247479014050, 2.73619237564129383521428007081, 4.03685302839075371552736363398, 4.76475503930049359765615416367, 5.85397472077304493409472420872, 7.09838679343894132356778277216, 7.70824339224273907816777791634, 8.679404920443158836926645411643, 9.426823526051934919784054973874, 10.48333582175580054215138384680