L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + 13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s − 0.999·27-s + (0.499 − 0.866i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)12-s + 13-s + (−0.499 + 0.866i)16-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s − 0.999·27-s + (0.499 − 0.866i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9743155287\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9743155287\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−11.07924151792112251502279409422, −10.16007332407537852246283458413, −9.297326645043563784764588132241, −8.788553050784162048287542945783, −7.912143297160387157663566225452, −6.29405059514878773586456937238, −5.36167476301944528068221235238, −4.66579447951206809222453269216, −3.43586964224177625454046849980, −1.95026617534710363294073473319,
1.51510240631486012590937124659, 3.25564162382170338486175477751, 3.95724525056339834796238152931, 5.38076380476054149900106785153, 6.79320098900243615942595236088, 7.46177816708211098543208681219, 8.333473784875796818150507635781, 8.824183963331549155982399384411, 10.04655475174635568119719741655, 11.16540106269567765907264127829