L(s) = 1 | + (−2.80 − 2.80i)2-s + 7.70i·4-s + (−25.0 + 25.0i)7-s + (−0.817 + 0.817i)8-s − 42.1i·11-s + (−25.7 − 25.7i)13-s + 140.·14-s + 66.2·16-s + (42.0 + 42.0i)17-s + 58.9i·19-s + (−118. + 118. i)22-s + (141. − 141. i)23-s + 144. i·26-s + (−192. − 192. i)28-s + 79.0·29-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.990i)2-s + 0.963i·4-s + (−1.35 + 1.35i)7-s + (−0.0361 + 0.0361i)8-s − 1.15i·11-s + (−0.548 − 0.548i)13-s + 2.67·14-s + 1.03·16-s + (0.600 + 0.600i)17-s + 0.711i·19-s + (−1.14 + 1.14i)22-s + (1.28 − 1.28i)23-s + 1.08i·26-s + (−1.30 − 1.30i)28-s + 0.506·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.582332 - 0.428943i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.582332 - 0.428943i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (2.80 + 2.80i)T + 8iT^{2} \) |
| 7 | \( 1 + (25.0 - 25.0i)T - 343iT^{2} \) |
| 11 | \( 1 + 42.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (25.7 + 25.7i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-42.0 - 42.0i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 58.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-141. + 141. i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 79.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-81.4 + 81.4i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 14.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-152. - 152. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (199. + 199. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-84.5 + 84.5i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 600.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-6.22 + 6.22i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 750. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-418. - 418. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 825. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-463. + 463. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 646.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-371. + 371. i)T - 9.12e5iT^{2} \) |
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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
−11.51961930261434860034617412718, −10.43220893297970040449279521874, −9.763163578487995319329790596883, −8.811110105692681318545038302671, −8.204712645829676038801131506304, −6.39154215354105009391667344366, −5.52607960085051991053771224545, −3.26441190361501990636957629807, −2.56843383081096901438580218821, −0.65417173935750980472939069444,
0.791619261880724189887916070344, 3.22436949288667685623341805560, 4.73484766524078939548687678111, 6.43103121003898502843423239099, 7.12533319999137863052949136831, 7.64593165121755641842792178043, 9.337831469340716717681540524531, 9.622028820124856316471446648137, 10.54294348418272957041038931891, 12.04194126128224600697976157456