| L(s) = 1 | + (0.810 + 1.15i)2-s + (−1.24 − 0.436i)3-s + (−0.687 + 1.87i)4-s + (1.45 + 0.332i)5-s + (−0.504 − 1.79i)6-s + (2.16 + 4.49i)7-s + (−2.73 + 0.724i)8-s + (−0.982 − 0.783i)9-s + (0.795 + 1.95i)10-s + (0.429 − 3.81i)11-s + (1.67 − 2.04i)12-s + (0.621 − 0.495i)13-s + (−3.45 + 6.14i)14-s + (−1.67 − 1.05i)15-s + (−3.05 − 2.58i)16-s + (2.22 − 2.22i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.819i)2-s + (−0.719 − 0.251i)3-s + (−0.343 + 0.939i)4-s + (0.651 + 0.148i)5-s + (−0.205 − 0.734i)6-s + (0.817 + 1.69i)7-s + (−0.966 + 0.256i)8-s + (−0.327 − 0.261i)9-s + (0.251 + 0.619i)10-s + (0.129 − 1.15i)11-s + (0.483 − 0.589i)12-s + (0.172 − 0.137i)13-s + (−0.923 + 1.64i)14-s + (−0.431 − 0.271i)15-s + (−0.763 − 0.645i)16-s + (0.539 − 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.914415 + 0.774356i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.914415 + 0.774356i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.810 - 1.15i)T \) |
| 29 | \( 1 + (-3.73 - 3.87i)T \) |
| good | 3 | \( 1 + (1.24 + 0.436i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.45 - 0.332i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-2.16 - 4.49i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.429 + 3.81i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.495i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.22 + 2.22i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.65 + 4.73i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (-3.02 + 0.689i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (4.84 - 3.04i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (2.57 - 0.290i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-0.977 - 0.977i)T + 41iT^{2} \) |
| 43 | \( 1 + (-5.24 + 8.34i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (7.11 + 0.801i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-1.32 + 5.81i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 0.973iT - 59T^{2} \) |
| 61 | \( 1 + (1.52 - 4.36i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (2.79 - 3.50i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-4.75 - 5.96i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (5.74 - 9.13i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-4.41 + 0.497i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (0.599 - 1.24i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.59 - 2.53i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (3.39 + 9.69i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−13.97297772045545590025370623552, −12.74350191368714435026036138441, −11.85350722434606567148303475778, −11.13153320722085222783511584514, −9.047046986615733118084237179417, −8.480867623463370389699326298420, −6.75052146100344785105412926697, −5.70067931003739202480444393357, −5.24060629088224894717474839414, −2.87295430756270247983179889728,
1.62157176214043520539554049360, 4.04185047338083802674417729041, 4.97805570307254183634715987378, 6.20955562243585502485748337381, 7.80414830259939474431833974401, 9.671070478945394504896585206504, 10.45797751784808903595069770342, 11.11060243369636311262826548348, 12.22493107703637621520152680522, 13.32259828281875619443590589484
