L(s) = 1 | + 1.54·3-s + (1.17 + 1.17i)7-s − 0.610·9-s + (−2.04 + 2.04i)11-s + 2.14i·13-s + (2.07 + 2.07i)17-s + (4.47 − 4.47i)19-s + (1.81 + 1.81i)21-s + (4.86 − 4.86i)23-s − 5.58·27-s + (5.51 + 5.51i)29-s + 5.72i·31-s + (−3.16 + 3.16i)33-s + 11.0i·37-s + 3.31i·39-s + ⋯ |
L(s) = 1 | + 0.892·3-s + (0.443 + 0.443i)7-s − 0.203·9-s + (−0.616 + 0.616i)11-s + 0.594i·13-s + (0.502 + 0.502i)17-s + (1.02 − 1.02i)19-s + (0.396 + 0.396i)21-s + (1.01 − 1.01i)23-s − 1.07·27-s + (1.02 + 1.02i)29-s + 1.02i·31-s + (−0.550 + 0.550i)33-s + 1.81i·37-s + 0.530i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.279858400\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.279858400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.54T + 3T^{2} \) |
| 7 | \( 1 + (-1.17 - 1.17i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.04 - 2.04i)T - 11iT^{2} \) |
| 13 | \( 1 - 2.14iT - 13T^{2} \) |
| 17 | \( 1 + (-2.07 - 2.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.47 + 4.47i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.86 + 4.86i)T - 23iT^{2} \) |
| 29 | \( 1 + (-5.51 - 5.51i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.72iT - 31T^{2} \) |
| 37 | \( 1 - 11.0iT - 37T^{2} \) |
| 41 | \( 1 - 11.4iT - 41T^{2} \) |
| 43 | \( 1 + 0.251iT - 43T^{2} \) |
| 47 | \( 1 + (-0.119 + 0.119i)T - 47iT^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + (-1.24 - 1.24i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.48 - 2.48i)T - 61iT^{2} \) |
| 67 | \( 1 + 9.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 + (7.85 + 7.85i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.86T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 3.63T + 89T^{2} \) |
| 97 | \( 1 + (9.89 + 9.89i)T + 97iT^{2} \) |
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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
−9.364819249284244867997545479336, −8.620109684661368157857393052791, −8.172812089988335776069528053088, −7.21081536354250993094965145847, −6.45057424631483194069087764265, −5.12834454373415385249275937948, −4.70568269714895085833597743237, −3.19169356974373412337600430988, −2.69141914187063312324347902766, −1.43742945793807177379903834831,
0.848646988715117127273962968312, 2.35685137412104699482381057327, 3.22501435657846594552084897346, 3.98286271486852243321235775348, 5.38223806830589462355219043462, 5.75441390489605761334980219971, 7.32152416580637650730266056564, 7.73976313313217369884720551324, 8.384709520325906026527314747964, 9.264718449163028708944011638744