L(s) = 1 | + (−2.16 + 1.72i)2-s + (0.581 − 1.63i)3-s + (1.25 − 5.51i)4-s + (2.91 + 1.98i)5-s + (1.55 + 4.53i)6-s + (−2.07 + 1.64i)7-s + (4.39 + 9.11i)8-s + (−2.32 − 1.89i)9-s + (−9.72 + 0.728i)10-s + (0.128 + 0.851i)11-s + (−8.26 − 5.25i)12-s + (0.596 + 3.95i)13-s + (1.64 − 7.13i)14-s + (4.93 − 3.59i)15-s + (−15.0 − 7.24i)16-s + (−1.51 + 1.40i)17-s + ⋯ |
L(s) = 1 | + (−1.52 + 1.22i)2-s + (0.335 − 0.942i)3-s + (0.629 − 2.75i)4-s + (1.30 + 0.887i)5-s + (0.636 + 1.85i)6-s + (−0.783 + 0.621i)7-s + (1.55 + 3.22i)8-s + (−0.774 − 0.632i)9-s + (−3.07 + 0.230i)10-s + (0.0387 + 0.256i)11-s + (−2.38 − 1.51i)12-s + (0.165 + 1.09i)13-s + (0.439 − 1.90i)14-s + (1.27 − 0.929i)15-s + (−3.76 − 1.81i)16-s + (−0.368 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.487875 + 0.604298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.487875 + 0.604298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.581 + 1.63i)T \) |
| 7 | \( 1 + (2.07 - 1.64i)T \) |
good | 2 | \( 1 + (2.16 - 1.72i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.91 - 1.98i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.128 - 0.851i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.596 - 3.95i)T + (-12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (1.51 - 1.40i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.72 - 0.995i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.379 - 1.23i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-6.03 - 6.50i)T + (-2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + 2.77iT - 31T^{2} \) |
| 37 | \( 1 + (-1.25 - 0.387i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.381 + 5.08i)T + (-40.5 - 6.11i)T^{2} \) |
| 43 | \( 1 + (0.323 + 4.31i)T + (-42.5 + 6.40i)T^{2} \) |
| 47 | \( 1 + (-2.93 - 3.67i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-3.02 - 9.81i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (3.49 + 1.68i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (3.17 - 0.723i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 + (0.0685 + 0.0156i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.591 - 3.92i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 - 8.89T + 79T^{2} \) |
| 83 | \( 1 + (-11.9 - 1.80i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (0.985 + 2.50i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-0.845 + 0.488i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.91960310123435047057077971386, −10.07557596694517480901895295595, −9.188881744596341879488842490045, −8.883853890477664302964434908712, −7.54663604842894958473435101020, −6.70911346475232406001491468724, −6.33356394886578343441295645418, −5.53777930912457835370295675794, −2.56573700722067912392559334943, −1.60964524659514122371890282182,
0.794999608805949313799312208780, 2.45696845518746486333831320145, 3.37890503124496108609003615174, 4.71347310185846710448101632200, 6.27637373448129478164722603651, 7.78730374700877228102972962064, 8.674835045940233457200071901855, 9.309144969795408144942133895610, 10.03511822113768438757650906314, 10.32999412698848875601928753711