| L(s) = 1 | − 3i·3-s − 5.86·5-s + (−18.3 + 2.64i)7-s − 9·9-s − 34.5·11-s − 55.8·13-s + 17.5i·15-s − 2.77i·17-s + 67.8i·19-s + (7.94 + 54.9i)21-s − 176. i·23-s − 90.6·25-s + 27i·27-s − 116. i·29-s − 312.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.524·5-s + (−0.989 + 0.143i)7-s − 0.333·9-s − 0.946·11-s − 1.19·13-s + 0.302i·15-s − 0.0395i·17-s + 0.819i·19-s + (0.0826 + 0.571i)21-s − 1.59i·23-s − 0.724·25-s + 0.192i·27-s − 0.743i·29-s − 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5011829996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5011829996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (18.3 - 2.64i)T \) |
| good | 5 | \( 1 + 5.86T + 125T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.77iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 67.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 176. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 116. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 280. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 15.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 6.34T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 288. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 475. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 473. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 796. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 877. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 33.8iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 700. iT - 9.12e5T^{2} \) |
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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
−9.262550703403102774811450437159, −8.187022821020480393627371656358, −7.65409671758681514613365869481, −6.83759329401667281602466580135, −6.00967823196915500652311630120, −5.10540784529690784660967990082, −3.98851029799380186430557427112, −2.91888426906625724847514430764, −2.11946353533486147906247462855, −0.38984051673185004435709086926,
0.27885305832697880015292943753, 2.22358872724057004533484705220, 3.26422628860115970846667242811, 3.96308469196115936587704329838, 5.14582625284237320544778463776, 5.67268026714027934924566729738, 7.13872321537664898442621896530, 7.38851511836546647845758303745, 8.546908412059968289690132743337, 9.489885933132121388421039929472
