L(s) = 1 | + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.728 + 0.684i)5-s + (−0.992 − 0.125i)8-s + (0.0627 − 0.998i)9-s + (0.968 − 0.248i)10-s + (−0.0235 + 0.374i)13-s + (−0.637 + 0.770i)16-s + (−0.0534 + 0.113i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (0.0627 + 0.998i)25-s + (0.303 + 0.220i)26-s + (−1.11 − 0.614i)29-s + (0.309 + 0.951i)32-s + ⋯ |
L(s) = 1 | + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.728 + 0.684i)5-s + (−0.992 − 0.125i)8-s + (0.0627 − 0.998i)9-s + (0.968 − 0.248i)10-s + (−0.0235 + 0.374i)13-s + (−0.637 + 0.770i)16-s + (−0.0534 + 0.113i)17-s + (−0.809 − 0.587i)18-s + (0.309 − 0.951i)20-s + (0.0627 + 0.998i)25-s + (0.303 + 0.220i)26-s + (−1.11 − 0.614i)29-s + (0.309 + 0.951i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.166454864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.166454864\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 5 | \( 1 + (-0.728 - 0.684i)T \) |
good | 3 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 13 | \( 1 + (0.0235 - 0.374i)T + (-0.992 - 0.125i)T^{2} \) |
| 17 | \( 1 + (0.0534 - 0.113i)T + (-0.637 - 0.770i)T^{2} \) |
| 19 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 23 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 29 | \( 1 + (1.11 + 0.614i)T + (0.535 + 0.844i)T^{2} \) |
| 31 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 37 | \( 1 + (1.23 - 1.49i)T + (-0.187 - 0.982i)T^{2} \) |
| 41 | \( 1 + (0.824 - 0.211i)T + (0.876 - 0.481i)T^{2} \) |
| 43 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 53 | \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \) |
| 59 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \) |
| 67 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 71 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 73 | \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \) |
| 79 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 89 | \( 1 + (0.996 - 0.394i)T + (0.728 - 0.684i)T^{2} \) |
| 97 | \( 1 + (1.62 + 0.895i)T + (0.535 + 0.844i)T^{2} \) |
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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
−11.07157968762670451324479966958, −10.09700018050733600707598749331, −9.569678467731536252880784216374, −8.644849482809304543003918616560, −6.98199558614614972620061179325, −6.23967281898917680231457191815, −5.30345272085402319967110629727, −3.96813481859337335533283745344, −3.01322906112589612014915589145, −1.72366068682744459228268545979,
2.19630064763167192328056236597, 3.77309123808479585519765328921, 5.09880463033174429559174564770, 5.44261134548909124210055955357, 6.67195022682726642402988522976, 7.65802075050363994791132049785, 8.509503992841381800037085604687, 9.306813459883023837002073208720, 10.34334426869112477897512205088, 11.41834176409250231797296079033