L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.707 − 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + (0.965 − 0.258i)6-s + (0.258 + 0.965i)7-s + 0.999·8-s + 1.00i·9-s + (−0.707 + 0.707i)10-s + (0.866 + 0.5i)11-s + (−0.258 + 0.965i)12-s + (−0.965 − 0.258i)13-s + (−0.965 − 0.258i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.707 − 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.965 + 0.258i)5-s + (0.965 − 0.258i)6-s + (0.258 + 0.965i)7-s + 0.999·8-s + 1.00i·9-s + (−0.707 + 0.707i)10-s + (0.866 + 0.5i)11-s + (−0.258 + 0.965i)12-s + (−0.965 − 0.258i)13-s + (−0.965 − 0.258i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7354123624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7354123624\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.965 - 0.258i)T \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
good | 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.869308647341484058927483093804, −9.263727612147364907572634515335, −8.355587764877922264653014102108, −7.42237909499062311709094404437, −6.73422195504853783797460487744, −5.99839193164497032384784285971, −5.39904185021796560795447954462, −4.57608802465618319602374162629, −2.41029028179913955011528016938, −1.53529201431609988316184500018,
0.896607296828147218097580884158, 2.26219625670319506252002425511, 3.66384378383109019841661187773, 4.48185493247106795587874301846, 5.18228394623557150861783510774, 6.49354693038846163832513519557, 7.13391649764318036819263343105, 8.547843839994472707059032845201, 9.175833827767757273033418836032, 9.814827876516995151394188229295