L(s) = 1 | + 2-s + 3-s + 4-s + 3.41·5-s + 6-s + 8-s + 9-s + 3.41·10-s + 2.41·11-s + 12-s − 13-s + 3.41·15-s + 16-s − 0.414·17-s + 18-s + 7.82·19-s + 3.41·20-s + 2.41·22-s − 1.41·23-s + 24-s + 6.65·25-s − 26-s + 27-s + 3.82·29-s + 3.41·30-s − 8.48·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.52·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.07·10-s + 0.727·11-s + 0.288·12-s − 0.277·13-s + 0.881·15-s + 0.250·16-s − 0.100·17-s + 0.235·18-s + 1.79·19-s + 0.763·20-s + 0.514·22-s − 0.294·23-s + 0.204·24-s + 1.33·25-s − 0.196·26-s + 0.192·27-s + 0.710·29-s + 0.623·30-s − 1.52·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.416028030\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.416028030\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 17 | \( 1 + 0.414T + 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 3.82T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 0.343T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 5.41T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.629243093350739789727097178171, −7.59293193685483276554368958481, −6.85734238982899370682010520285, −6.25915914814804895108321497983, −5.35639526990505694452046802057, −4.93987988213078821040913320483, −3.68884533360026957461492748793, −3.05177542792737903772090459892, −2.02952332092683050589231995322, −1.38885018051565769559417429259,
1.38885018051565769559417429259, 2.02952332092683050589231995322, 3.05177542792737903772090459892, 3.68884533360026957461492748793, 4.93987988213078821040913320483, 5.35639526990505694452046802057, 6.25915914814804895108321497983, 6.85734238982899370682010520285, 7.59293193685483276554368958481, 8.629243093350739789727097178171