| L(s) = 1 | + (0.0797 + 0.0149i)2-s + (0.284 − 3.06i)3-s + (−1.85 − 0.720i)4-s + (−1.51 + 2.00i)5-s + (0.0683 − 0.240i)6-s + (3.32 − 2.05i)7-s + (−0.275 − 0.170i)8-s + (−6.38 − 1.19i)9-s + (−0.150 + 0.137i)10-s + (0.349 + 0.0653i)11-s + (−2.73 + 5.49i)12-s + (3.77 − 2.33i)13-s + (0.295 − 0.114i)14-s + (5.71 + 5.21i)15-s + (2.92 + 2.66i)16-s + (0.150 + 0.529i)17-s + ⋯ |
| L(s) = 1 | + (0.0563 + 0.0105i)2-s + (0.164 − 1.77i)3-s + (−0.929 − 0.360i)4-s + (−0.676 + 0.896i)5-s + (0.0279 − 0.0981i)6-s + (1.25 − 0.777i)7-s + (−0.0973 − 0.0602i)8-s + (−2.12 − 0.397i)9-s + (−0.0476 + 0.0434i)10-s + (0.105 + 0.0197i)11-s + (−0.790 + 1.58i)12-s + (1.04 − 0.648i)13-s + (0.0789 − 0.0305i)14-s + (1.47 + 1.34i)15-s + (0.731 + 0.667i)16-s + (0.0365 + 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.581032 - 0.695418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581032 - 0.695418i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (0.0812 + 10.1i)T \) |
| good | 2 | \( 1 + (-0.0797 - 0.0149i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.284 + 3.06i)T + (-2.94 - 0.551i)T^{2} \) |
| 5 | \( 1 + (1.51 - 2.00i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (-3.32 + 2.05i)T + (3.12 - 6.26i)T^{2} \) |
| 11 | \( 1 + (-0.349 - 0.0653i)T + (10.2 + 3.97i)T^{2} \) |
| 13 | \( 1 + (-3.77 + 2.33i)T + (5.79 - 11.6i)T^{2} \) |
| 17 | \( 1 + (-0.150 - 0.529i)T + (-14.4 + 8.94i)T^{2} \) |
| 19 | \( 1 + (-0.346 - 3.73i)T + (-18.6 + 3.49i)T^{2} \) |
| 23 | \( 1 + (-6.69 + 1.25i)T + (21.4 - 8.30i)T^{2} \) |
| 29 | \( 1 + (5.02 - 6.65i)T + (-7.93 - 27.8i)T^{2} \) |
| 31 | \( 1 + (5.55 + 5.06i)T + (2.86 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.571 + 1.14i)T + (-22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (1.87 + 2.47i)T + (-11.2 + 39.4i)T^{2} \) |
| 43 | \( 1 + (-0.141 - 0.284i)T + (-25.9 + 34.3i)T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 + (-0.180 - 1.94i)T + (-52.0 + 9.73i)T^{2} \) |
| 59 | \( 1 + (-9.69 - 6.00i)T + (26.2 + 52.8i)T^{2} \) |
| 61 | \( 1 + (-1.96 - 6.89i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (-0.111 - 0.0689i)T + (29.8 + 59.9i)T^{2} \) |
| 71 | \( 1 + (-2.38 - 3.15i)T + (-19.4 + 68.2i)T^{2} \) |
| 73 | \( 1 + (-1.67 - 2.21i)T + (-19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (1.39 - 1.84i)T + (-21.6 - 75.9i)T^{2} \) |
| 83 | \( 1 + (3.29 - 2.03i)T + (36.9 - 74.2i)T^{2} \) |
| 89 | \( 1 + (-4.97 + 1.92i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (0.155 - 0.548i)T + (-82.4 - 51.0i)T^{2} \) |
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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
−13.46161463833778920974197345501, −12.80197053059000670250081394703, −11.39019682991700353637290627671, −10.73040607730535093553938182268, −8.679857821656811387359716352106, −7.83759889870811241818413095564, −7.03784415819958476609571817029, −5.56981384718562693097175765894, −3.62600682720448312680972022297, −1.28468390147185094086462379705,
3.63643805919971163631816040756, 4.69237013099882677955918811174, 5.22588369465593516280585848867, 8.194698951690084025314380887148, 8.821149810722318026827003003941, 9.413331843133579662116598298962, 11.10937086464012699744471184801, 11.68105775285214566530610018604, 13.15973711470955491285169671190, 14.38351180235573552009906272835
