| L(s) = 1 | + (0.0411 + 0.0587i)3-s + (−2.20 + 0.365i)5-s + (0.553 + 0.148i)7-s + (1.02 − 2.81i)9-s + (2.49 − 4.32i)11-s + (4.89 + 3.42i)13-s + (−0.112 − 0.114i)15-s + (1.10 − 2.36i)17-s + (3.41 + 2.71i)19-s + (0.0140 + 0.0386i)21-s + (4.15 + 0.363i)23-s + (4.73 − 1.61i)25-s + (0.415 − 0.111i)27-s + (−5.28 − 1.92i)29-s + (−5.80 + 3.34i)31-s + ⋯ |
| L(s) = 1 | + (0.0237 + 0.0339i)3-s + (−0.986 + 0.163i)5-s + (0.209 + 0.0560i)7-s + (0.341 − 0.938i)9-s + (0.752 − 1.30i)11-s + (1.35 + 0.951i)13-s + (−0.0289 − 0.0295i)15-s + (0.267 − 0.572i)17-s + (0.782 + 0.622i)19-s + (0.00306 + 0.00842i)21-s + (0.867 + 0.0758i)23-s + (0.946 − 0.322i)25-s + (0.0799 − 0.0214i)27-s + (−0.982 − 0.357i)29-s + (−1.04 + 0.601i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28110 - 0.271373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28110 - 0.271373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.20 - 0.365i)T \) |
| 19 | \( 1 + (-3.41 - 2.71i)T \) |
| good | 3 | \( 1 + (-0.0411 - 0.0587i)T + (-1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.553 - 0.148i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.49 + 4.32i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.89 - 3.42i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 2.36i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-4.15 - 0.363i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (5.28 + 1.92i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.80 - 3.34i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.89 + 5.89i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.05 + 0.186i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.00412 - 0.0471i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-10.8 + 5.04i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.268 - 3.07i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-6.31 + 2.30i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (6.37 - 5.35i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.22 - 11.2i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (0.975 - 1.16i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.22 - 1.55i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-1.81 + 10.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.33 - 4.98i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (1.70 + 9.66i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (10.0 + 4.69i)T + (62.3 + 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.54149838532215704004104751642, −10.60977526746480723900627959253, −9.068377083702289169177014557347, −8.809004780476692342461878229241, −7.46777062598780559729793476076, −6.63252585029511313963544463946, −5.54677873565143897142308708818, −3.85862067192461532050147270617, −3.50605793347761346571494852637, −1.10882055229684084518377683632,
1.47908971922763505846424140321, 3.37887519990427926281840565744, 4.42394305844526600743088633395, 5.42666939866934793699136815201, 6.96123833000538073487758372378, 7.65393025690227398951208801358, 8.517455220173328723173074616171, 9.549309526791724040017457710948, 10.78464196469463616729534160678, 11.22237986852515504044336212042
