L(s) = 1 | + (−0.195 + 0.980i)2-s + (−0.923 − 0.382i)4-s + (−0.555 − 0.831i)7-s + (0.555 − 0.831i)8-s + (−0.831 − 0.555i)9-s + (−1.36 − 0.728i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (0.980 − 1.19i)22-s + (−0.324 − 1.63i)23-s + (−0.980 − 0.195i)25-s + (0.195 + 0.980i)28-s + (−0.0924 − 0.172i)29-s + (−0.831 + 0.555i)32-s + ⋯ |
L(s) = 1 | + (−0.195 + 0.980i)2-s + (−0.923 − 0.382i)4-s + (−0.555 − 0.831i)7-s + (0.555 − 0.831i)8-s + (−0.831 − 0.555i)9-s + (−1.36 − 0.728i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (0.980 − 1.19i)22-s + (−0.324 − 1.63i)23-s + (−0.980 − 0.195i)25-s + (0.195 + 0.980i)28-s + (−0.0924 − 0.172i)29-s + (−0.831 + 0.555i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4371379363\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4371379363\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.195 - 0.980i)T \) |
| 7 | \( 1 + (0.555 + 0.831i)T \) |
good | 3 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 5 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 11 | \( 1 + (1.36 + 0.728i)T + (0.555 + 0.831i)T^{2} \) |
| 13 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 19 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 23 | \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (0.0924 + 0.172i)T + (-0.555 + 0.831i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1.11 - 1.36i)T + (-0.195 + 0.980i)T^{2} \) |
| 41 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (-0.273 - 0.902i)T + (-0.831 + 0.555i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 53 | \( 1 + (-0.902 + 1.68i)T + (-0.555 - 0.831i)T^{2} \) |
| 59 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 61 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 67 | \( 1 + (0.555 + 0.168i)T + (0.831 + 0.555i)T^{2} \) |
| 71 | \( 1 + (-0.636 + 0.425i)T + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.750 + 1.81i)T + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 89 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.10397112503661538089813049744, −9.222222793876138940235595339063, −8.207624296630967467988758965754, −7.83388728059061853692833521210, −6.56329711689591331039020199463, −6.09499150419981572512658551783, −5.06325356737193829185885937116, −3.98466839131066052686796422158, −2.85731327921526230397213453086, −0.43712567007360193083461238717,
2.11760821690747758599519481532, 2.77909637376431326714344090739, 3.96196216786550395570674062040, 5.33239601196896075119524131563, 5.67683742242700816561408902097, 7.46128048255211379862778245258, 8.037694345844274758496289883445, 9.060751756139679194036383665139, 9.662941169418501451574267796059, 10.48961438167336879760818706418