L(s) = 1 | + (−0.891 − 0.453i)2-s + (−0.453 − 0.891i)3-s + (0.587 + 0.809i)4-s + 1.00i·6-s + (−0.156 − 0.987i)8-s + (−0.587 + 0.809i)9-s + (1.65 + 0.398i)11-s + (0.453 − 0.891i)12-s + (−0.309 + 0.951i)16-s + (−0.652 + 0.399i)17-s + (0.891 − 0.453i)18-s + (0.987 − 1.15i)19-s + (−1.29 − 1.10i)22-s + (−0.809 + 0.587i)24-s + (0.951 + 0.309i)25-s + ⋯ |
L(s) = 1 | + (−0.891 − 0.453i)2-s + (−0.453 − 0.891i)3-s + (0.587 + 0.809i)4-s + 1.00i·6-s + (−0.156 − 0.987i)8-s + (−0.587 + 0.809i)9-s + (1.65 + 0.398i)11-s + (0.453 − 0.891i)12-s + (−0.309 + 0.951i)16-s + (−0.652 + 0.399i)17-s + (0.891 − 0.453i)18-s + (0.987 − 1.15i)19-s + (−1.29 − 1.10i)22-s + (−0.809 + 0.587i)24-s + (0.951 + 0.309i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6330762072\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6330762072\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 3 | \( 1 + (0.453 + 0.891i)T \) |
| 41 | \( 1 + (0.987 + 0.156i)T \) |
good | 5 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + (0.987 - 0.156i)T^{2} \) |
| 11 | \( 1 + (-1.65 - 0.398i)T + (0.891 + 0.453i)T^{2} \) |
| 13 | \( 1 + (0.156 - 0.987i)T^{2} \) |
| 17 | \( 1 + (0.652 - 0.399i)T + (0.453 - 0.891i)T^{2} \) |
| 19 | \( 1 + (-0.987 + 1.15i)T + (-0.156 - 0.987i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.453 - 0.891i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \) |
| 47 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 53 | \( 1 + (0.453 + 0.891i)T^{2} \) |
| 59 | \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 67 | \( 1 + (1.89 - 0.453i)T + (0.891 - 0.453i)T^{2} \) |
| 71 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 73 | \( 1 + (-0.437 - 0.437i)T + iT^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + 1.61iT - T^{2} \) |
| 89 | \( 1 + (-0.465 + 0.0366i)T + (0.987 - 0.156i)T^{2} \) |
| 97 | \( 1 + (0.303 + 1.26i)T + (-0.891 + 0.453i)T^{2} \) |
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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
−10.04875110062468456073004616763, −9.013260988684300129915558923358, −8.644289364581255708585142157358, −7.31441201234347721585396430522, −6.96610039066645090259519138957, −6.13569429934206231106121836848, −4.74604920996282944379553561651, −3.44772913983657342925313501484, −2.16438609730424965040166821238, −1.09758725210947538873308542636,
1.25885570119708666740550869740, 3.10821430325557371430690832880, 4.28407347074361029509437148813, 5.33109484467350937084898211212, 6.25590554407147558698300931544, 6.80721780170545165401626796139, 8.043454709584610781048564336064, 8.935558177101693854414324453188, 9.446834570280179022582679648563, 10.15047450554914734900456056210