L(s) = 1 | + 2.44i·5-s − 2i·7-s − 4.89·11-s − 3.46·13-s + 4.24i·17-s − 6.92i·19-s + 8.48·23-s − 0.999·25-s − 7.34i·29-s − 2i·31-s + 4.89·35-s + 6.92·37-s − 4.24i·41-s + 8.48·47-s + 3·49-s + ⋯ |
L(s) = 1 | + 1.09i·5-s − 0.755i·7-s − 1.47·11-s − 0.960·13-s + 1.02i·17-s − 1.58i·19-s + 1.76·23-s − 0.199·25-s − 1.36i·29-s − 0.359i·31-s + 0.828·35-s + 1.13·37-s − 0.662i·41-s + 1.23·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.338229532\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.338229532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 7.34iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−8.961623323325754656783527953584, −7.935073317299652542900023641020, −7.27953079792128623761746605006, −6.87992732386413826412997061247, −5.81496849766687185895284745291, −4.91561426751245306642362596942, −4.08628868895568455074876651354, −2.82515306134225521471330752746, −2.47104672096324767291579253445, −0.55319703954386230615386950087,
0.984039087276358375965743924064, 2.35762053796601978388804338902, 3.11550007072910139082090421664, 4.54716554111462629705362511920, 5.27932809546136194452674365925, 5.49291254358024889680324333385, 6.88411946267151944462611276372, 7.66552384358328991394738359717, 8.358095848669956077016769995075, 9.061113038141601277710633035777