L(s) = 1 | + (1.80 − 0.286i)2-s + (−0.665 + 1.30i)3-s + (−0.624 + 0.202i)4-s + (−3.20 − 3.83i)5-s + (−0.828 + 2.54i)6-s + (3.62 − 3.62i)7-s + (−7.58 + 3.86i)8-s + (4.02 + 5.54i)9-s + (−6.88 − 6.01i)10-s + (5.24 + 3.81i)11-s + (0.150 − 0.950i)12-s + (−4.11 − 0.652i)13-s + (5.51 − 7.59i)14-s + (7.14 − 1.63i)15-s + (−10.4 + 7.60i)16-s + (−6.15 − 12.0i)17-s + ⋯ |
L(s) = 1 | + (0.902 − 0.143i)2-s + (−0.221 + 0.435i)3-s + (−0.156 + 0.0507i)4-s + (−0.641 − 0.767i)5-s + (−0.138 + 0.424i)6-s + (0.518 − 0.518i)7-s + (−0.948 + 0.483i)8-s + (0.447 + 0.615i)9-s + (−0.688 − 0.601i)10-s + (0.477 + 0.346i)11-s + (0.0125 − 0.0791i)12-s + (−0.316 − 0.0501i)13-s + (0.394 − 0.542i)14-s + (0.476 − 0.109i)15-s + (−0.654 + 0.475i)16-s + (−0.362 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14198 + 0.00598721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14198 + 0.00598721i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (3.20 + 3.83i)T \) |
good | 2 | \( 1 + (-1.80 + 0.286i)T + (3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (0.665 - 1.30i)T + (-5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-3.62 + 3.62i)T - 49iT^{2} \) |
| 11 | \( 1 + (-5.24 - 3.81i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (4.11 + 0.652i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (6.15 + 12.0i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-25.5 - 8.31i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (5.03 + 31.7i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (52.2 - 16.9i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-8.09 + 24.9i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (6.89 - 43.5i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-29.6 + 21.5i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-28.0 - 28.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.78 - 4.98i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-17.0 + 33.4i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (14.1 + 19.5i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (34.1 + 24.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (2.47 + 4.85i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-33.2 - 102. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (14.9 + 94.1i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-106. + 34.5i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (54.8 - 27.9i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (6.88 - 9.48i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-46.3 - 23.6i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−17.12747602218474726023082157142, −16.09577173937169662617943114773, −14.76284636100908074120339174997, −13.56800084047349742260656970797, −12.38190092528299994688710287109, −11.25950841624422122488025554630, −9.409407289956557379769096309347, −7.71501518161671282566705888702, −5.09744292930555446529635678178, −4.14946297240286931536427581240,
3.78300917578973074114260101496, 5.78502284590704948206784317060, 7.31191577457309662854847401822, 9.320725966431287427747517043659, 11.41233656574962009174748007800, 12.30956134026680244789254602310, 13.71504157392852381818200968830, 14.86856919922266951243023951003, 15.60746121291258188162770597234, 17.70097162384220940441220102471