L(s) = 1 | + 1.45i·3-s + 3.69i·5-s − 3.33i·7-s + 0.878·9-s − 4.75i·11-s + 2.51·13-s − 5.38·15-s + (−3.76 + 1.68i)17-s − 0.982·19-s + 4.85·21-s − 4.02i·23-s − 8.66·25-s + 5.64i·27-s − 0.375i·29-s − 3.17i·31-s + ⋯ |
L(s) = 1 | + 0.840i·3-s + 1.65i·5-s − 1.25i·7-s + 0.292·9-s − 1.43i·11-s + 0.697·13-s − 1.39·15-s + (−0.912 + 0.408i)17-s − 0.225·19-s + 1.05·21-s − 0.838i·23-s − 1.73·25-s + 1.08i·27-s − 0.0697i·29-s − 0.569i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.931199135\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.931199135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.76 - 1.68i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 1.45iT - 3T^{2} \) |
| 5 | \( 1 - 3.69iT - 5T^{2} \) |
| 7 | \( 1 + 3.33iT - 7T^{2} \) |
| 11 | \( 1 + 4.75iT - 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 19 | \( 1 + 0.982T + 19T^{2} \) |
| 23 | \( 1 + 4.02iT - 23T^{2} \) |
| 29 | \( 1 + 0.375iT - 29T^{2} \) |
| 31 | \( 1 + 3.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.30iT - 37T^{2} \) |
| 41 | \( 1 + 7.88iT - 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 61 | \( 1 + 0.836iT - 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 0.00541iT - 73T^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.561211069468642448054304721958, −7.60708894198013953773511829422, −7.00508527093760097499316208025, −6.34261743972278132955423760770, −5.71036970052559371349247086582, −4.30953257365725299393811219239, −3.91173405936831256315403186235, −3.27521105364934138351578395649, −2.28037654573196896791962186363, −0.68519926261003701776835020573,
0.973206440309139802348748040909, 1.81505796783244289359801277336, 2.46141907131773455655316549540, 4.07817410042761620161415012387, 4.66574269679204514499064785785, 5.43322586229670254040685506765, 6.11682137388286041468926087458, 7.02682252754842599825939022738, 7.70742405489992174253212152431, 8.448293016262634729388638136802