L(s) = 1 | + (0.496 − 0.286i)2-s + 3-s + (−0.835 + 1.44i)4-s + (2.74 + 1.58i)5-s + (0.496 − 0.286i)6-s + (−2.25 − 1.38i)7-s + 2.10i·8-s + 9-s + 1.81·10-s − 0.352i·11-s + (−0.835 + 1.44i)12-s + (1.81 + 3.11i)13-s + (−1.51 − 0.0401i)14-s + (2.74 + 1.58i)15-s + (−1.06 − 1.85i)16-s + (0.444 − 0.769i)17-s + ⋯ |
L(s) = 1 | + (0.350 − 0.202i)2-s + 0.577·3-s + (−0.417 + 0.723i)4-s + (1.22 + 0.708i)5-s + (0.202 − 0.116i)6-s + (−0.852 − 0.522i)7-s + 0.743i·8-s + 0.333·9-s + 0.574·10-s − 0.106i·11-s + (−0.241 + 0.417i)12-s + (0.502 + 0.864i)13-s + (−0.405 − 0.0107i)14-s + (0.708 + 0.408i)15-s + (−0.267 − 0.462i)16-s + (0.107 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75139 + 0.519534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75139 + 0.519534i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (2.25 + 1.38i)T \) |
| 13 | \( 1 + (-1.81 - 3.11i)T \) |
good | 2 | \( 1 + (-0.496 + 0.286i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.74 - 1.58i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 0.352iT - 11T^{2} \) |
| 17 | \( 1 + (-0.444 + 0.769i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 1.53iT - 19T^{2} \) |
| 23 | \( 1 + (2.26 + 3.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.213 + 0.370i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.47 + 4.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.88 + 1.66i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.73 + 2.73i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.380 + 0.659i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.53 + 4.92i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 3.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.96 + 5.75i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 9.35iT - 67T^{2} \) |
| 71 | \( 1 + (-10.7 + 6.22i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.88 + 3.40i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.48 - 6.03i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 + (5.29 - 3.05i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 6.56i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.20715527564843747957375422791, −10.99424018555290841761430814913, −9.935632924783398754357824699005, −9.295614305854918446595922913417, −8.217359143012621503506624756533, −6.94182897485576309710657262333, −6.13644362038266654729897543611, −4.48825958917929063905953610542, −3.33953094833281569475742882466, −2.33155668306679981468047417833,
1.51216228249455923976110459960, 3.22166073979782383726714873319, 4.80570628112892852897038460007, 5.80359968844544523832942076667, 6.41547597386109093907862721336, 8.172040609528483048258615239456, 9.179539102878545136885348116421, 9.764877472294731570005584747923, 10.43152705828263272248253225845, 12.22050685455362582376693472189