L(s) = 1 | − 5-s − 0.669i·7-s + 6.53·11-s − 1.67·13-s + 4.86·17-s + 7.93i·19-s + (−2.87 + 3.83i)23-s + 25-s + 5.55i·29-s − 2.03·31-s + 0.669i·35-s − 0.940i·37-s + 2.80i·41-s − 6.91i·43-s + 2.23i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.253i·7-s + 1.96·11-s − 0.465·13-s + 1.18·17-s + 1.82i·19-s + (−0.600 + 0.799i)23-s + 0.200·25-s + 1.03i·29-s − 0.364·31-s + 0.113i·35-s − 0.154i·37-s + 0.437i·41-s − 1.05i·43-s + 0.326i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.700396688\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.700396688\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + (2.87 - 3.83i)T \) |
good | 7 | \( 1 + 0.669iT - 7T^{2} \) |
| 11 | \( 1 - 6.53T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 - 4.86T + 17T^{2} \) |
| 19 | \( 1 - 7.93iT - 19T^{2} \) |
| 29 | \( 1 - 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 0.940iT - 37T^{2} \) |
| 41 | \( 1 - 2.80iT - 41T^{2} \) |
| 43 | \( 1 + 6.91iT - 43T^{2} \) |
| 47 | \( 1 - 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 2.74iT - 59T^{2} \) |
| 61 | \( 1 + 1.34iT - 61T^{2} \) |
| 67 | \( 1 - 2.66iT - 67T^{2} \) |
| 71 | \( 1 + 4.57iT - 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 13.7iT - 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 2.14iT - 97T^{2} \) |
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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
−7.888664294518741579201494928433, −7.32096894016965983264054307546, −6.68009080574882365521268733204, −5.87271771687984910334666956297, −5.32381372400671106134847679926, −4.11189402493097215334146078795, −3.85125618872600086975340957686, −3.11910156043375954844450892258, −1.69524728610587807999222732716, −1.16846937645347658839097133672,
0.43414633008990605579866840612, 1.42823220227031093664338122803, 2.51820789939249704702620638989, 3.32265988767272976264433307769, 4.19096020611732824730581850633, 4.62654625156711142222360830659, 5.64109742982134940758159639588, 6.33505921961810412284101795795, 6.96571359178897786169713386143, 7.53406091166679610195272184885