L(s) = 1 | + (2.50 − 0.396i)2-s + (0.910 + 1.78i)3-s + (4.21 − 1.36i)4-s + (−0.509 + 2.17i)5-s + (2.98 + 4.11i)6-s + (−1.61 − 0.820i)7-s + (5.48 − 2.79i)8-s + (−0.599 + 0.824i)9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (−0.352 − 2.22i)13-s + (−4.35 − 1.41i)14-s + (−4.35 + 1.07i)15-s + (5.46 − 3.96i)16-s + (0.526 − 3.32i)17-s + (−1.17 + 2.30i)18-s + ⋯ |
L(s) = 1 | + (1.77 − 0.280i)2-s + (0.525 + 1.03i)3-s + (2.10 − 0.684i)4-s + (−0.227 + 0.973i)5-s + (1.21 + 1.67i)6-s + (−0.608 − 0.310i)7-s + (1.93 − 0.987i)8-s + (−0.199 + 0.274i)9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (−0.0978 − 0.617i)13-s + (−1.16 − 0.378i)14-s + (−1.12 + 0.276i)15-s + (1.36 − 0.991i)16-s + (0.127 − 0.805i)17-s + (−0.276 + 0.542i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.08724 + 1.40350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.08724 + 1.40350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.509 - 2.17i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-2.50 + 0.396i)T + (1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.910 - 1.78i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (1.61 + 0.820i)T + (4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (0.352 + 2.22i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.526 + 3.32i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.403 - 1.24i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.16 - 3.16i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.907 + 2.79i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.24 - 2.35i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 2.85i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 0.402i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.55 + 4.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.88 + 3.50i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (6.29 - 0.996i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (11.7 - 3.82i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.47 - 7.53i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.46 + 2.46i)T + 67iT^{2} \) |
| 71 | \( 1 + (-5.47 + 3.98i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.28 + 8.41i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.926 - 0.673i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 1.63i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (-0.121 - 0.768i)T + (-92.2 + 29.9i)T^{2} \) |
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
−10.72474906502933203816790897126, −10.24366291639294940259456534863, −9.398001940024634704803678829333, −7.80176926487027416429282902160, −6.84296626224439864783329500801, −5.98478949049281703005184283164, −4.91324913029649291159358416344, −3.85820471266634196256850108449, −3.38692450285055822812297481901, −2.50884943768192289949043770204,
1.74478697295595390261424468046, 2.85624755710038237452217468730, 4.06640973550908042599239149877, 4.89770601828264763052560135023, 6.07830250536019434059812821746, 6.65448920772449448457133696634, 7.70819531815603647306133311958, 8.410964431876519878353056392268, 9.575430527205579032344536935629, 11.04641773984093906664200720718