| L(s) = 1 | + (1.83 − 1.28i)5-s − 3.76i·7-s − 3.54·11-s − 1.00i·13-s + 1.56i·17-s − 7.21·19-s − 6.18i·23-s + (1.71 − 4.69i)25-s + 3·29-s − 5.78·31-s + (−4.83 − 6.90i)35-s + 0.851i·37-s − 3.11·41-s + 2.70i·43-s + 9.74i·47-s + ⋯ |
| L(s) = 1 | + (0.819 − 0.573i)5-s − 1.42i·7-s − 1.06·11-s − 0.278i·13-s + 0.378i·17-s − 1.65·19-s − 1.28i·23-s + (0.342 − 0.939i)25-s + 0.557·29-s − 1.03·31-s + (−0.816 − 1.16i)35-s + 0.139i·37-s − 0.487·41-s + 0.412i·43-s + 1.42i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166514991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166514991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.83 + 1.28i)T \) |
| good | 7 | \( 1 + 3.76iT - 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 + 1.00iT - 13T^{2} \) |
| 17 | \( 1 - 1.56iT - 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 6.18iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 0.851iT - 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 2.70iT - 43T^{2} \) |
| 47 | \( 1 - 9.74iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 + 2.91iT - 67T^{2} \) |
| 71 | \( 1 - 7.21T + 71T^{2} \) |
| 73 | \( 1 + 16.7iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 + 3.86iT - 97T^{2} \) |
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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
−9.020224289299495146633728087955, −8.289444929027238244758276958184, −7.55704190738873293107806243714, −6.57093031215336881986410635424, −5.89370124301001679012283220094, −4.74623316190789654542311940349, −4.26304822690885536934897831696, −2.88913319805570886551325435049, −1.75728708931563229761306017033, −0.40961284485859985984533296879,
2.01030207605220562837040614207, 2.48353746806309346876229104173, 3.62551841075756196207728543359, 5.14609064625159213449900165663, 5.53267496956680947402083318046, 6.42992573299776805143497800149, 7.20091229306815449313799266672, 8.326673054345742412300324565809, 8.894414380694111964237976383703, 9.733842466178367633172766833341
