L(s) = 1 | + (−0.544 + 1.19i)3-s + (0.928 − 0.371i)4-s + (−0.0913 − 0.0268i)5-s + (−0.468 − 0.540i)9-s + (0.235 + 0.971i)11-s + (−0.0623 + 1.30i)12-s + (0.0816 − 0.0941i)15-s + (0.723 − 0.690i)16-s + (−0.0947 + 0.00904i)20-s + (0.481 + 0.676i)23-s + (−0.833 − 0.535i)25-s + (−0.357 + 0.105i)27-s + (0.581 + 0.299i)31-s + (−1.28 − 0.247i)33-s + (−0.635 − 0.327i)36-s + (0.327 − 0.566i)37-s + ⋯ |
L(s) = 1 | + (−0.544 + 1.19i)3-s + (0.928 − 0.371i)4-s + (−0.0913 − 0.0268i)5-s + (−0.468 − 0.540i)9-s + (0.235 + 0.971i)11-s + (−0.0623 + 1.30i)12-s + (0.0816 − 0.0941i)15-s + (0.723 − 0.690i)16-s + (−0.0947 + 0.00904i)20-s + (0.481 + 0.676i)23-s + (−0.833 − 0.535i)25-s + (−0.357 + 0.105i)27-s + (0.581 + 0.299i)31-s + (−1.28 − 0.247i)33-s + (−0.635 − 0.327i)36-s + (0.327 − 0.566i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9823678878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9823678878\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.235 - 0.971i)T \) |
| 67 | \( 1 + (-0.580 + 0.814i)T \) |
good | 2 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 3 | \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 5 | \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \) |
| 7 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 13 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 17 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 19 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 23 | \( 1 + (-0.481 - 0.676i)T + (-0.327 + 0.945i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.581 - 0.299i)T + (0.580 + 0.814i)T^{2} \) |
| 37 | \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (1.95 - 0.186i)T + (0.981 - 0.189i)T^{2} \) |
| 53 | \( 1 + (0.0671 - 0.466i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.396 + 0.254i)T + (0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 71 | \( 1 + (-1.56 + 0.625i)T + (0.723 - 0.690i)T^{2} \) |
| 73 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 79 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 83 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 89 | \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.74305884371968549052729737936, −9.834439975986727054027697177472, −9.580841209620206995853548058096, −8.104382403482254590001208401821, −7.12365376138732040791346841077, −6.23640732068394546012211088591, −5.27612898039559191653060412696, −4.49104151456441491017116237019, −3.35064095550687568269257423549, −1.88235187336754045450457269563,
1.29279331669749068583501158313, 2.56707178261188949432709710975, 3.72994676618597513089481595673, 5.37591932014226898205869492904, 6.41588656561212148481949100205, 6.69563152567766572790447732092, 7.81255113088588178259808108418, 8.292668703412235535617862844531, 9.635278576107895198939185976135, 10.80984170413097547046601542096