L(s) = 1 | + (0.258 − 0.965i)3-s + (0.258 + 0.965i)7-s + (−0.866 − 0.499i)9-s + 21-s + (1.67 + 0.448i)23-s + (−0.707 + 0.707i)27-s + (0.866 − 1.5i)29-s + (1.5 − 0.866i)41-s + (−1.93 + 0.517i)43-s + (1.67 − 0.448i)47-s + (0.5 − 0.866i)61-s + (0.258 − 0.965i)63-s + (0.965 + 0.258i)67-s + (0.866 − 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (0.258 + 0.965i)7-s + (−0.866 − 0.499i)9-s + 21-s + (1.67 + 0.448i)23-s + (−0.707 + 0.707i)27-s + (0.866 − 1.5i)29-s + (1.5 − 0.866i)41-s + (−1.93 + 0.517i)43-s + (1.67 − 0.448i)47-s + (0.5 − 0.866i)61-s + (0.258 − 0.965i)63-s + (0.965 + 0.258i)67-s + (0.866 − 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.443300816\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.443300816\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
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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
−8.547953908587324940413858068247, −7.986420971799502323774164008937, −7.15148578307249902086654288441, −6.51568175243000774570027600371, −5.68200726930464547364507863348, −5.09511535639075341659869383387, −3.88379050519461418042499571716, −2.79334281540399881339795367190, −2.23974975576718639749561806506, −1.03367402490081284037359720645,
1.13812322417201503930187416497, 2.64360837480741990100940618805, 3.38478544607371818413645973307, 4.29265145039428555527401189451, 4.84162502119233901552783854661, 5.61712528360994364551685714092, 6.73230625915502514091636166104, 7.32586620562079326157485376928, 8.245200107699561298374067054875, 8.843001075753144302337380564163