L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (−0.500 − 0.866i)10-s + (−0.5 − 0.133i)11-s + (0.965 − 0.258i)13-s + (0.866 + 0.499i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)18-s + (0.133 + 0.5i)19-s + (−0.965 + 0.258i)20-s + (−0.258 + 0.448i)22-s + (1.22 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)9-s + (−0.500 − 0.866i)10-s + (−0.5 − 0.133i)11-s + (0.965 − 0.258i)13-s + (0.866 + 0.499i)14-s + (0.500 + 0.866i)16-s + (−0.707 − 0.707i)18-s + (0.133 + 0.5i)19-s + (−0.965 + 0.258i)20-s + (−0.258 + 0.448i)22-s + (1.22 − 0.707i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.519932774\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.519932774\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
| 13 | \( 1 + (-0.965 + 0.258i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 53 | \( 1 - 1.93T + T^{2} \) |
| 59 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
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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
−8.771736962243243444305958929226, −8.175186389032033021396732264176, −6.69779573890155749463257189676, −6.00672992022464363157585316555, −5.38894926971155445177019572671, −4.68028176169847057521577023589, −3.64705263703107618027171609912, −2.89552136647385113067248482373, −1.89891832648132920006832312945, −0.924097181072261621246002069276,
1.39431365246747759526358760827, 2.83159638876134164824187614153, 3.61612429187789009733044586817, 4.59185927492115475570705042606, 5.22928772414832363930304507785, 6.14255331534017355867032709104, 6.78308729051174833265377978275, 7.36453622487441332635287992725, 7.911532863718658468260832695241, 8.896099548647967323891607048958