L(s) = 1 | + (1.26 + 0.639i)2-s + (−1.57 + 0.730i)3-s + (1.18 + 1.61i)4-s + (−2.44 − 0.0838i)6-s − 1.25i·7-s + (0.458 + 2.79i)8-s + (1.93 − 2.29i)9-s + 3.02i·11-s + (−3.03 − 1.67i)12-s + 5.65i·13-s + (0.803 − 1.58i)14-s + (−1.20 + 3.81i)16-s + 2.45i·17-s + (3.90 − 1.65i)18-s + 1.77·19-s + ⋯ |
L(s) = 1 | + (0.891 + 0.452i)2-s + (−0.906 + 0.421i)3-s + (0.590 + 0.806i)4-s + (−0.999 − 0.0342i)6-s − 0.474i·7-s + (0.162 + 0.986i)8-s + (0.644 − 0.764i)9-s + 0.911i·11-s + (−0.875 − 0.482i)12-s + 1.56i·13-s + (0.214 − 0.423i)14-s + (−0.301 + 0.953i)16-s + 0.595i·17-s + (0.920 − 0.390i)18-s + 0.407·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796835 + 1.50667i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796835 + 1.50667i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.26 - 0.639i)T \) |
| 3 | \( 1 + (1.57 - 0.730i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.25iT - 7T^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 2.45iT - 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 6.45iT - 37T^{2} \) |
| 41 | \( 1 - 7.57iT - 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 4.91iT - 59T^{2} \) |
| 61 | \( 1 + 8.16iT - 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 + 7.36iT - 79T^{2} \) |
| 83 | \( 1 - 15.7iT - 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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
−11.20036885190727126594028571721, −10.21702435866547118671863871958, −9.403461712162288525285339473974, −8.040462858868052953852382707673, −7.02196054951105601511241057935, −6.41121483948325059358284528646, −5.44814110183745743162331884816, −4.28518659019456313602019743511, −3.99017907904863459638097683700, −1.99984488708176989817587907090,
0.806599038141925397845604054509, 2.42958457463274481999725669530, 3.61825492950917663399770126958, 5.05156247559900836637636690979, 5.63050223105704425381209803310, 6.37048125913690896187565399357, 7.47340558930175411880448241036, 8.504938849319441136512360754042, 10.07838675950376927870038631715, 10.45438495101082672059275537200