L(s) = 1 | − 1.21i·5-s + (0.355 + 2.62i)7-s − 4.06·11-s + 1.42·13-s − 4.74·17-s + 4.84·19-s + 6.03i·23-s + 3.51·25-s + 3.45·29-s − 3.15i·31-s + (3.19 − 0.432i)35-s − 9.24i·37-s − 1.24·41-s + 3.23i·43-s − 9.43·47-s + ⋯ |
L(s) = 1 | − 0.544i·5-s + (0.134 + 0.990i)7-s − 1.22·11-s + 0.396·13-s − 1.15·17-s + 1.11·19-s + 1.25i·23-s + 0.703·25-s + 0.641·29-s − 0.566i·31-s + (0.539 − 0.0731i)35-s − 1.51i·37-s − 0.194·41-s + 0.492i·43-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7156761345\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7156761345\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.355 - 2.62i)T \) |
good | 5 | \( 1 + 1.21iT - 5T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 6.03iT - 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 + 3.15iT - 31T^{2} \) |
| 37 | \( 1 + 9.24iT - 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 3.23iT - 43T^{2} \) |
| 47 | \( 1 + 9.43T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 - 10.2iT - 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 6.51iT - 67T^{2} \) |
| 71 | \( 1 + 2.14iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−8.729388845582524222329346815872, −8.056524207853598188276378432830, −7.40857928270663556332965302296, −6.43407216180417229845505616113, −5.53919332438505556325932300747, −5.17780546573334993610509068836, −4.30207734054890601191603804250, −3.13519583483793395219393974733, −2.41534853532699426321003090311, −1.32461426870905301107033044843,
0.20431802685430528534078507872, 1.52256379973918493395621768986, 2.79795104413622439064701885427, 3.30622807904011291777175784392, 4.59382512321609255969505222160, 4.88753076896210147339445907913, 6.11916659452136785638935713785, 6.79934900665202867267573145936, 7.32109820167600032743712606648, 8.209294079932775187011614860648