| L(s) = 1 | + 2.44i·3-s − 1.41·7-s − 2.99·9-s + 2i·11-s − 5.65i·13-s − 4.89·17-s − 6i·19-s − 3.46i·21-s − 7.07·23-s − 6.92i·29-s + 6.92·31-s − 4.89·33-s − 2.82i·37-s + 13.8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + 1.41i·3-s − 0.534·7-s − 0.999·9-s + 0.603i·11-s − 1.56i·13-s − 1.18·17-s − 1.37i·19-s − 0.755i·21-s − 1.47·23-s − 1.28i·29-s + 1.24·31-s − 0.852·33-s − 0.464i·37-s + 2.21·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5172095470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5172095470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44iT - 3T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 2.44iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.525354478390503594778481109386, −8.594520851693606716328667439405, −7.82635512401877166142607587319, −6.69476363974916607857770605120, −5.87099834231982517677791951883, −4.85365292531088433936258617778, −4.32088405571878303056742700573, −3.31083414108393185761599492123, −2.40980621806028837572283963758, −0.19257897509366683125021025519,
1.46046292971988945974909376639, 2.23555568025589576347109645376, 3.51199236891614315933432793329, 4.52904517939079956026351000222, 5.89802126704323465245372142352, 6.53461458947478891858498039951, 6.92914746972738752922129580987, 8.043287350142916544730995764349, 8.525026094393861908876952832186, 9.479453226758772664343544907124
