L(s) = 1 | − 2-s + 3-s + 4-s − 2.27·5-s − 6-s − 8-s + 9-s + 2.27·10-s − 11-s + 12-s − 4.63·13-s − 2.27·15-s + 16-s + 0.554·17-s − 18-s + 4.07·19-s − 2.27·20-s + 22-s − 6.93·23-s − 24-s + 0.171·25-s + 4.63·26-s + 27-s − 2.82·29-s + 2.27·30-s + 7.68·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.01·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.719·10-s − 0.301·11-s + 0.288·12-s − 1.28·13-s − 0.587·15-s + 0.250·16-s + 0.134·17-s − 0.235·18-s + 0.935·19-s − 0.508·20-s + 0.213·22-s − 1.44·23-s − 0.204·24-s + 0.0343·25-s + 0.908·26-s + 0.192·27-s − 0.525·29-s + 0.415·30-s + 1.38·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.024890486\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024890486\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2.27T + 5T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 - 0.554T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.554T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.440T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 6.51T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−8.452515085050972437626052074546, −7.84121980611326255526904529210, −7.58422360360580858824430502948, −6.72337897311139557230854412891, −5.68653489664922849437200161513, −4.65796789384715268447361979317, −3.85465032870054956556928133328, −2.91026403855153755392812581941, −2.10516790966525592416541477660, −0.63489052297660975331385629687,
0.63489052297660975331385629687, 2.10516790966525592416541477660, 2.91026403855153755392812581941, 3.85465032870054956556928133328, 4.65796789384715268447361979317, 5.68653489664922849437200161513, 6.72337897311139557230854412891, 7.58422360360580858824430502948, 7.84121980611326255526904529210, 8.452515085050972437626052074546