L(s) = 1 | + (1 − i)2-s + (−0.258 + 0.965i)3-s − i·4-s + (0.258 − 0.965i)5-s + (0.707 + 1.22i)6-s + (−0.866 − 0.499i)9-s + (−0.707 − 1.22i)10-s + (0.5 − 0.866i)11-s + (0.965 + 0.258i)12-s + (−0.965 + 0.258i)13-s + (0.866 + 0.499i)15-s + 16-s + (0.965 + 0.258i)17-s + (−1.36 + 0.366i)18-s + (−0.965 − 0.258i)20-s + ⋯ |
L(s) = 1 | + (1 − i)2-s + (−0.258 + 0.965i)3-s − i·4-s + (0.258 − 0.965i)5-s + (0.707 + 1.22i)6-s + (−0.866 − 0.499i)9-s + (−0.707 − 1.22i)10-s + (0.5 − 0.866i)11-s + (0.965 + 0.258i)12-s + (−0.965 + 0.258i)13-s + (0.866 + 0.499i)15-s + 16-s + (0.965 + 0.258i)17-s + (−1.36 + 0.366i)18-s + (−0.965 − 0.258i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.934328903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934328903\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.214612722496675344571582856809, −8.647819439538231726096468671029, −7.69328187773805384578870747406, −6.16934815207759303890968047820, −5.63989828796484322828509404075, −4.69048698007818462412823120419, −4.40937512678355418206405012528, −3.37997756610512774342707511563, −2.58099363900888673709128282976, −1.14768537852026998382749829543,
1.63262765987102489895582455518, 2.81259148558304148635532166441, 3.74233808638759047755027115793, 4.99210954448189681650351262669, 5.54059431176278378670896615080, 6.34921950879596254248529429917, 6.98743827826351495211925271403, 7.48804475300307048859165135093, 7.970383546673881320757613929824, 9.514951481612265132168358350416