L(s) = 1 | + (−0.766 + 0.642i)3-s + (0.326 + 0.118i)7-s + (0.173 − 0.984i)9-s + (0.0603 + 0.342i)13-s + (0.766 − 0.642i)19-s + (−0.326 + 0.118i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + 1.87·31-s + (−0.5 + 0.866i)37-s + (−0.266 − 0.223i)39-s − 1.53·43-s + (−0.673 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (0.347 + 1.96i)61-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)3-s + (0.326 + 0.118i)7-s + (0.173 − 0.984i)9-s + (0.0603 + 0.342i)13-s + (0.766 − 0.642i)19-s + (−0.326 + 0.118i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + 1.87·31-s + (−0.5 + 0.866i)37-s + (−0.266 − 0.223i)39-s − 1.53·43-s + (−0.673 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (0.347 + 1.96i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9357549465\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9357549465\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 19 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 1.87T + T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + 1.53T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.725328947129475770946023633768, −8.878799901707911724782085591299, −8.130513670940421167901125886592, −6.92798667606457959783570419870, −6.45845050171312786699878803435, −5.25714073048020324393843875045, −4.89114152407306294171599276201, −3.83631381006523274946573803409, −2.82491444931338758442373668423, −1.20184479162547336071927553370,
0.988834784132603097322303742704, 2.19397373532623993479713177154, 3.43801705970172223722121514541, 4.73177750292028140206438868564, 5.28803122714954705012308388124, 6.32361581746902540820410277509, 6.83620545041115128733250492783, 7.973341282203490709762331016857, 8.192374008328661921017912116481, 9.512324080454904254890053133193