| L(s) = 1 | + 0.732i·5-s − 7-s − 4.19i·11-s + 1.26·17-s + 7.46i·19-s − 8.73·23-s + 4.46·25-s − 3.46i·29-s + 4.92·31-s − 0.732i·35-s − 10i·37-s − 1.26·41-s + 0.928i·43-s + 6.92·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.327i·5-s − 0.377·7-s − 1.26i·11-s + 0.307·17-s + 1.71i·19-s − 1.82·23-s + 0.892·25-s − 0.643i·29-s + 0.885·31-s − 0.123i·35-s − 1.64i·37-s − 0.198·41-s + 0.141i·43-s + 1.01·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313617699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313617699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 4.19iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 0.928iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.46iT - 61T^{2} \) |
| 67 | \( 1 + 7.46iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 + 8.39iT - 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.151075768390921020468262733184, −7.76160274635338484146320089632, −6.71391832087432059729426121990, −5.90015065226944711634362097233, −5.70440004506850664796171822565, −4.30174136580657657011241273236, −3.63725387314756800121681071508, −2.87019133457855893607214931238, −1.77444296562930855913048072626, −0.42258028664382903963560359691,
1.04859275969819145266774259185, 2.24156970388928448736207447019, 3.03962282337937458567069250130, 4.22560862067560772269646216458, 4.73057092403956957505865346557, 5.56353371920335253501831718267, 6.57291689475018898890531187644, 7.02650390402903793490928681667, 7.85798750630017251710515743137, 8.657602872607839985239711026466
