L(s) = 1 | + (0.273 − 1.89i)3-s + (−0.888 + 0.458i)4-s + (−0.271 − 0.595i)5-s + (−2.57 − 0.755i)9-s + (−0.995 − 0.0950i)11-s + (0.627 + 1.81i)12-s + (−1.20 + 0.353i)15-s + (0.580 − 0.814i)16-s + (0.514 + 0.404i)20-s + (−0.264 + 0.105i)23-s + (0.374 − 0.432i)25-s + (−1.34 + 2.93i)27-s + (1.42 − 0.273i)31-s + (−0.452 + 1.86i)33-s + (2.63 − 0.507i)36-s + (−0.723 − 1.25i)37-s + ⋯ |
L(s) = 1 | + (0.273 − 1.89i)3-s + (−0.888 + 0.458i)4-s + (−0.271 − 0.595i)5-s + (−2.57 − 0.755i)9-s + (−0.995 − 0.0950i)11-s + (0.627 + 1.81i)12-s + (−1.20 + 0.353i)15-s + (0.580 − 0.814i)16-s + (0.514 + 0.404i)20-s + (−0.264 + 0.105i)23-s + (0.374 − 0.432i)25-s + (−1.34 + 2.93i)27-s + (1.42 − 0.273i)31-s + (−0.452 + 1.86i)33-s + (2.63 − 0.507i)36-s + (−0.723 − 1.25i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6069281213\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6069281213\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.995 + 0.0950i)T \) |
| 67 | \( 1 + (-0.928 - 0.371i)T \) |
good | 2 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 3 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \) |
| 7 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 13 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 17 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 19 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 23 | \( 1 + (0.264 - 0.105i)T + (0.723 - 0.690i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \) |
| 37 | \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 43 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 + (0.370 + 0.291i)T + (0.235 + 0.971i)T^{2} \) |
| 53 | \( 1 + (1.67 + 1.07i)T + (0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-1.30 - 1.50i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 71 | \( 1 + (-1.16 + 0.600i)T + (0.580 - 0.814i)T^{2} \) |
| 73 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 79 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 83 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 89 | \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.05628898354393303982773847336, −8.839904421622808212433839315819, −8.327873673817914781675140151782, −7.78536159822136676133039614219, −6.92998431524217817426142758718, −5.80190457505834798012033394032, −4.85768295179950800322367240887, −3.40172539331668539823056400032, −2.26080977152802948409920279808, −0.63608141169059796707986662926,
2.82390744651427554007395587825, 3.67768641965168193184073725535, 4.72355944442024107502830023252, 5.16609910291080863634452033425, 6.30680291203111815521450685959, 7.981410962688335866415814480323, 8.563250581683822228265082270335, 9.579721012519320878291022514358, 10.01473204922227460330628084238, 10.72939702782407210496494352517