| L(s) = 1 | + (1.41 + 1.41i)5-s + 7-s + (−0.748 + 0.748i)11-s + (4.24 + 4.24i)13-s − 7.50i·17-s − 4.59i·23-s − 0.999i·25-s + (5.26 − 5.26i)29-s − 8.49i·31-s + (1.41 + 1.41i)35-s + (−1 + i)37-s + 0.978·41-s + (−6.80 − 6.80i)43-s + 6.36·47-s + 49-s + ⋯ |
| L(s) = 1 | + (0.632 + 0.632i)5-s + 0.377·7-s + (−0.225 + 0.225i)11-s + (1.17 + 1.17i)13-s − 1.82i·17-s − 0.958i·23-s − 0.199i·25-s + (0.976 − 0.976i)29-s − 1.52i·31-s + (0.239 + 0.239i)35-s + (−0.164 + 0.164i)37-s + 0.152·41-s + (−1.03 − 1.03i)43-s + 0.927·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417067752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417067752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.748 - 0.748i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.24 - 4.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.50iT - 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 + 4.59iT - 23T^{2} \) |
| 29 | \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.49iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.978T + 41T^{2} \) |
| 43 | \( 1 + (6.80 + 6.80i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.36T + 47T^{2} \) |
| 53 | \( 1 + (-8.17 - 8.17i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.51 + 4.51i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.249 - 0.249i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.750 - 0.750i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.62iT - 71T^{2} \) |
| 73 | \( 1 - 8.61iT - 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 + (2.47 + 2.47i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.55T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.513020027070983776477873599944, −7.61697403792473534570918574593, −6.82302389633843479927097054072, −6.34916392715841392769562146130, −5.51711757445111696426224863450, −4.59457307193095252612534822286, −3.92275373314478051215856124432, −2.63833819045190154589830660746, −2.18326882655671578243834125359, −0.800350933295364142105159723779,
1.13099388196479686736547314742, 1.69829262432842716030304100617, 3.11213250476238636941249938256, 3.76131035948508827424452122261, 4.88578566758667592006925942302, 5.55824916976684943663803678136, 6.02959249925505392230791091203, 6.96114589931313532895384954428, 7.980199654255919094528702184777, 8.539352072427915793986433163448
