| L(s) = 1 | + (0.292 + 1.70i)3-s + (2 − 2i)7-s + (−2.82 + i)9-s + 5.65i·11-s + (2.82 + 2.82i)17-s + 4i·19-s + (4 + 2.82i)21-s + (4.24 − 4.24i)23-s + (−2.53 − 4.53i)27-s − 5.65·29-s + 8·31-s + (−9.65 + 1.65i)33-s + (−8 + 8i)37-s + 5.65i·41-s + (2 + 2i)43-s + ⋯ |
| L(s) = 1 | + (0.169 + 0.985i)3-s + (0.755 − 0.755i)7-s + (−0.942 + 0.333i)9-s + 1.70i·11-s + (0.685 + 0.685i)17-s + 0.917i·19-s + (0.872 + 0.617i)21-s + (0.884 − 0.884i)23-s + (−0.487 − 0.872i)27-s − 1.05·29-s + 1.43·31-s + (−1.68 + 0.288i)33-s + (−1.31 + 1.31i)37-s + 0.883i·41-s + (0.304 + 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15136 + 1.08221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15136 + 1.08221i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 - 1.70i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (8 - 8i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2 - 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.41 + 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.65 + 5.65i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-6 + 6i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (8 + 8i)T + 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (-9.89 + 9.89i)T - 83iT^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-8 + 8i)T - 97iT^{2} \) |
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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
−10.45190105737124379120231919022, −10.26790997470103556179578276889, −9.288907941500785219884959084649, −8.198736528586330265477450939756, −7.54434022681102129259711095366, −6.32615977138206328877332237972, −4.93579949982330244563366460651, −4.47885609832175514594065076086, −3.34470924901840740743919297413, −1.75838974038615239535563466949,
0.936459502804013404569416563802, 2.43534831154197271873436270643, 3.42898663251707303857578393591, 5.27487929781584120190708097856, 5.77765085315488845289309526320, 6.99183340137787615935079542034, 7.81375719714500017868336676411, 8.727355781855999880679092779502, 9.148108900018671106456004147899, 10.76474368085959758832814518758
