L(s) = 1 | + (−1.33 − 1.49i)2-s + (−0.446 + 3.97i)4-s + 6.56i·7-s + (6.52 − 4.63i)8-s + 2.26i·11-s − 14.8·13-s + (9.79 − 8.75i)14-s + (−15.6 − 3.55i)16-s − 26.8·17-s − 10.8i·19-s + (3.38 − 3.02i)22-s − 36.4i·23-s + (19.8 + 22.1i)26-s + (−26.1 − 2.93i)28-s + 35.2·29-s + ⋯ |
L(s) = 1 | + (−0.666 − 0.745i)2-s + (−0.111 + 0.993i)4-s + 0.938i·7-s + (0.815 − 0.579i)8-s + 0.206i·11-s − 1.14·13-s + (0.699 − 0.625i)14-s + (−0.975 − 0.221i)16-s − 1.57·17-s − 0.572i·19-s + (0.153 − 0.137i)22-s − 1.58i·23-s + (0.762 + 0.853i)26-s + (−0.932 − 0.104i)28-s + 1.21·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8255676751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8255676751\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 + 1.49i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 6.56iT - 49T^{2} \) |
| 11 | \( 1 - 2.26iT - 121T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 + 26.8T + 289T^{2} \) |
| 19 | \( 1 + 10.8iT - 361T^{2} \) |
| 23 | \( 1 + 36.4iT - 529T^{2} \) |
| 29 | \( 1 - 35.2T + 841T^{2} \) |
| 31 | \( 1 - 23.8iT - 961T^{2} \) |
| 37 | \( 1 - 54.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 23.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 51.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 6.92iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 107.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 111. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 31.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 59.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 142. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.14T + 7.92e3T^{2} \) |
| 97 | \( 1 + 126.T + 9.40e3T^{2} \) |
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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
−9.705079677449647565361003477707, −8.812365125061460561052212735987, −8.439576064562597341556712231159, −7.19199481735006972053911789777, −6.50491520118772868643397772918, −5.00389739373704294820689684279, −4.26421795459622298190178445210, −2.65816166632715292459209130482, −2.25936852607721271612535858736, −0.41278118102436942929658473554,
0.964516918168743166581048030846, 2.40958328255955667496056137631, 4.11386087778541214438847963591, 4.90466638397752031422341139591, 6.05663963724421069499171099151, 6.85298196166054204163062344297, 7.64404566008307354968245358544, 8.243084013167317693076488493775, 9.565296620868155017665937546947, 9.703199918231180952063612866897