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 | \( 1 - 13.8T + 97T^{2} \) |
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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
−10.45438495101082672059275537200, −10.07838675950376927870038631715, −8.504938849319441136512360754042, −7.47340558930175411880448241036, −6.37048125913690896187565399357, −5.63050223105704425381209803310, −5.05156247559900836637636690979, −3.61825492950917663399770126958, −2.42958457463274481999725669530, −0.806599038141925397845604054509,
1.99984488708176989817587907090, 3.99017907904863459638097683700, 4.28518659019456313602019743511, 5.44814110183745743162331884816, 6.41121483948325059358284528646, 7.02196054951105601511241057935, 8.040462858868052953852382707673, 9.403461712162288525285339473974, 10.21702435866547118671863871958, 11.20036885190727126594028571721