L(s) = 1 | + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + 2.73·7-s + (−2 − 1.99i)8-s + 2i·11-s − 3.46i·13-s + (1 + 3.73i)14-s + (1.99 − 3.46i)16-s − 3.46·17-s + 7.46i·19-s + (−2.73 + 0.732i)22-s + 4.19·23-s + (4.73 − 1.26i)26-s + (−4.73 + 2.73i)28-s + 6.92i·29-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + 1.03·7-s + (−0.707 − 0.707i)8-s + 0.603i·11-s − 0.960i·13-s + (0.267 + 0.997i)14-s + (0.499 − 0.866i)16-s − 0.840·17-s + 1.71i·19-s + (−0.582 + 0.156i)22-s + 0.874·23-s + (0.928 − 0.248i)26-s + (−0.894 + 0.516i)28-s + 1.28i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.759314178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.759314178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 - 8.73iT - 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 4.92iT - 61T^{2} \) |
| 67 | \( 1 + 7.26iT - 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.329178622575976287238617960733, −8.545290947626753631773761318318, −7.88319925034561733633239607497, −7.32895478578751269488575577706, −6.35342789167998192351429089688, −5.50413984072621867437478269186, −4.81287414242273894467219978595, −4.03650263376282594148551657355, −2.87826714235681184658344459995, −1.34199150637179295573846830704,
0.65892818651162965730428811272, 1.97631472798150239608124480479, 2.76062461967093825512111621067, 4.08047426656114933265909162256, 4.64226078198880089883451875933, 5.45422510116094619496675468821, 6.50110993518538510628342355347, 7.44961894939055142553507036800, 8.666814812371835171078460023961, 8.873971492019650506102202208267