L(s) = 1 | + (0.866 + 0.5i)4-s + (−0.258 − 0.965i)7-s + (1.22 − 1.22i)13-s + (0.499 + 0.866i)16-s + (−0.866 + 0.5i)19-s + (0.258 − 0.965i)28-s + (1 − 1.73i)31-s + (−0.448 + 1.67i)37-s + (−0.866 + 0.499i)49-s + (1.67 − 0.448i)52-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (−1.67 + 0.448i)67-s + (0.448 + 1.67i)73-s − 0.999·76-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)4-s + (−0.258 − 0.965i)7-s + (1.22 − 1.22i)13-s + (0.499 + 0.866i)16-s + (−0.866 + 0.5i)19-s + (0.258 − 0.965i)28-s + (1 − 1.73i)31-s + (−0.448 + 1.67i)37-s + (−0.866 + 0.499i)49-s + (1.67 − 0.448i)52-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (−1.67 + 0.448i)67-s + (0.448 + 1.67i)73-s − 0.999·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.368376007\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368376007\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
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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
−9.850464409587151526666868244190, −8.413750585109308632822974950472, −8.099575795113733301581011906152, −7.19785284439916028999226136836, −6.37741764039721931278476214888, −5.79836078031074469992515593169, −4.32921940695173903882516054259, −3.57347278219466344129823894733, −2.70076653172242960124420367632, −1.27172041915631068316538179779,
1.57703436320421469760729459561, 2.46631360104468861330299454859, 3.55187887649718642116361517769, 4.77412651392797161115933520152, 5.74527013954070106563302180287, 6.47490970590098677799816789934, 6.90793487519500525510843984242, 8.178575683522602377414505933345, 8.937303605969508275711146433460, 9.501611436166530566118598453356