L(s) = 1 | + 2i·7-s + 30·11-s + 4i·13-s + 90i·17-s + 28·19-s − 120i·23-s − 210·29-s − 4·31-s + 200i·37-s + 240·41-s + 136i·43-s − 120i·47-s + 339·49-s + 30i·53-s + 450·59-s + ⋯ |
L(s) = 1 | + 0.107i·7-s + 0.822·11-s + 0.0853i·13-s + 1.28i·17-s + 0.338·19-s − 1.08i·23-s − 1.34·29-s − 0.0231·31-s + 0.888i·37-s + 0.914·41-s + 0.482i·43-s − 0.372i·47-s + 0.988·49-s + 0.0777i·53-s + 0.992·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.876204273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.876204273\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 - 30T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 90iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 210T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 200iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 240T + 6.89e4T^{2} \) |
| 43 | \( 1 - 136iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 120iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 30iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 450T + 2.05e5T^{2} \) |
| 61 | \( 1 + 166T + 2.26e5T^{2} \) |
| 67 | \( 1 - 908iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 250iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 916T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 420T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3iT - 9.12e5T^{2} \) |
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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
−9.862048838099089684281713303064, −8.984757123786402427642232785285, −8.303989356516527336828144194491, −7.29274695838758324888561995994, −6.39292657444229191763917115297, −5.63442383737230546666706575152, −4.40374737342277227871344860888, −3.61601024255644277325011327017, −2.28149160136605896831341064411, −1.08753953328986409167968713206,
0.55303527308519681293374018256, 1.85831384695908548445332944375, 3.18367518960115744679590450596, 4.11666844502765768121624199982, 5.22743037597371884508436182095, 6.05734782291819883945001197779, 7.21412527715072580097826691702, 7.64521413829382144285755038282, 9.077458509640400681824568308323, 9.327477117999820591859985928513