| 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
−17.70097162384220940441220102471, −15.60746121291258188162770597234, −14.86856919922266951243023951003, −13.71504157392852381818200968830, −12.30956134026680244789254602310, −11.41233656574962009174748007800, −9.320725966431287427747517043659, −7.31191577457309662854847401822, −5.78502284590704948206784317060, −3.78300917578973074114260101496,
4.14946297240286931536427581240, 5.09744292930555446529635678178, 7.71501518161671282566705888702, 9.409407289956557379769096309347, 11.25950841624422122488025554630, 12.38190092528299994688710287109, 13.56800084047349742260656970797, 14.76284636100908074120339174997, 16.09577173937169662617943114773, 17.12747602218474726023082157142
