L(s) = 1 | + (−1.73 + i)2-s − 1.73·3-s + (1.99 − 3.46i)4-s + (2.99 − 1.73i)6-s − 6.92·7-s + 7.99i·8-s + 2.99·9-s + 6.92i·11-s + (−3.46 + 5.99i)12-s + 2i·13-s + (11.9 − 6.92i)14-s + (−8 − 13.8i)16-s − 10i·17-s + (−5.19 + 2.99i)18-s − 20.7i·19-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s − 0.577·3-s + (0.499 − 0.866i)4-s + (0.499 − 0.288i)6-s − 0.989·7-s + 0.999i·8-s + 0.333·9-s + 0.629i·11-s + (−0.288 + 0.499i)12-s + 0.153i·13-s + (0.857 − 0.494i)14-s + (−0.5 − 0.866i)16-s − 0.588i·17-s + (−0.288 + 0.166i)18-s − 1.09i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.712064 - 0.0213509i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712064 - 0.0213509i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 6.92T + 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 + 10iT - 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 - 27.7T + 529T^{2} \) |
| 29 | \( 1 - 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 + 2iT - 9.40e3T^{2} \) |
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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
−11.26377481698931038262569360627, −10.46392818735690328953319130411, −9.510596002810014356498458949680, −8.912854128886238747873658719338, −7.35630990999148080948590064674, −6.83844828745120090773025301813, −5.78846829533214390619808945907, −4.63868614555646558522736893729, −2.64476146287557353020140330191, −0.68259592614840950207321827264,
0.948206505104185409518595159602, 2.83176954909788888508846346584, 4.01000301823539361850927910742, 5.82590736967970447174524390888, 6.68049722837820779770038869804, 7.79944600821344634124289252830, 8.859621281122429876540628699039, 9.782511555570929039445553029867, 10.58190631892630427384023711463, 11.30316550269371386423415760544