L(s) = 1 | + 2.08·2-s − 3-s + 2.35·4-s + 2.43·5-s − 2.08·6-s + 1.35·7-s + 0.734·8-s + 9-s + 5.08·10-s + 11-s − 2.35·12-s − 13-s + 2.82·14-s − 2.43·15-s − 3.17·16-s − 3.08·17-s + 2.08·18-s + 2.64·19-s + 5.73·20-s − 1.35·21-s + 2.08·22-s + 1.52·23-s − 0.734·24-s + 0.944·25-s − 2.08·26-s − 27-s + 3.17·28-s + ⋯ |
L(s) = 1 | + 1.47·2-s − 0.577·3-s + 1.17·4-s + 1.09·5-s − 0.851·6-s + 0.510·7-s + 0.259·8-s + 0.333·9-s + 1.60·10-s + 0.301·11-s − 0.678·12-s − 0.277·13-s + 0.753·14-s − 0.629·15-s − 0.793·16-s − 0.748·17-s + 0.491·18-s + 0.607·19-s + 1.28·20-s − 0.295·21-s + 0.444·22-s + 0.317·23-s − 0.149·24-s + 0.188·25-s − 0.409·26-s − 0.192·27-s + 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.915934926\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.915934926\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 + 0.876T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 - 0.703T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−11.54937662271450704965666716103, −10.46567326119404736863330725672, −9.595448558731147708462874337757, −8.397485363757343844217123364261, −6.76539568446533992033310151987, −6.31103040513898922387995084903, −5.11651818650083728177403132581, −4.75588118468598547262284162829, −3.22621431945199266546246563057, −1.84763668785801946375743417955,
1.84763668785801946375743417955, 3.22621431945199266546246563057, 4.75588118468598547262284162829, 5.11651818650083728177403132581, 6.31103040513898922387995084903, 6.76539568446533992033310151987, 8.397485363757343844217123364261, 9.595448558731147708462874337757, 10.46567326119404736863330725672, 11.54937662271450704965666716103