L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)5-s + 8-s − 0.999·10-s + (0.5 + 0.866i)16-s − 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)34-s + (−0.5 − 0.866i)38-s + (−0.500 + 0.866i)40-s + 0.999·46-s + (−1 − 1.73i)47-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)5-s + 8-s − 0.999·10-s + (0.5 + 0.866i)16-s − 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)34-s + (−0.5 − 0.866i)38-s + (−0.500 + 0.866i)40-s + 0.999·46-s + (−1 − 1.73i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.055880458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055880458\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−11.46669758665371225287517015186, −10.82659253995829168080045524512, −10.02975199629696553070240282299, −8.609836530904373440437456688220, −7.70797103589107901473095663846, −6.70128422669499857385229090308, −6.28696348819327222079394110170, −4.88534069693732128461046620751, −3.96283284314292897956239151632, −2.36470940730744213209832599340,
1.72241182277389574123742867895, 3.20070404887720902396814416450, 4.30746236260083919008629593266, 5.01996500504280180830195282942, 6.57761387172952325273896913430, 7.73355615850760394401301738471, 8.567094889802317980942082513128, 9.569657504300470819616103727247, 10.78043409833624124403047556995, 11.37914445742390888364191252848