L(s) = 1 | + 1.62·2-s + 0.654·4-s − 2.04·5-s − 3.70·7-s − 2.19·8-s − 3.33·10-s − 11-s − 5.86·13-s − 6.04·14-s − 4.88·16-s − 0.842·17-s + 1.22·19-s − 1.33·20-s − 1.62·22-s + 8.32·23-s − 0.817·25-s − 9.56·26-s − 2.42·28-s − 4.26·29-s − 7.99·31-s − 3.56·32-s − 1.37·34-s + 7.58·35-s + 2.78·37-s + 1.99·38-s + 4.48·40-s − 8.34·41-s + ⋯ |
L(s) = 1 | + 1.15·2-s + 0.327·4-s − 0.914·5-s − 1.40·7-s − 0.774·8-s − 1.05·10-s − 0.301·11-s − 1.62·13-s − 1.61·14-s − 1.22·16-s − 0.204·17-s + 0.281·19-s − 0.299·20-s − 0.347·22-s + 1.73·23-s − 0.163·25-s − 1.87·26-s − 0.459·28-s − 0.791·29-s − 1.43·31-s − 0.631·32-s − 0.235·34-s + 1.28·35-s + 0.458·37-s + 0.324·38-s + 0.708·40-s − 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8253803225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8253803225\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 + 0.842T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 - 8.32T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 67 | \( 1 + 0.830T + 67T^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 8.50T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73603498573659811230938579823, −7.24613236490352677519154150787, −6.64577003571098300904118715846, −5.73543405007254104994403528215, −5.12443958063506590635176336211, −4.42137501373968687167020578592, −3.61943960348352280076080732345, −3.10806833600470773382331709745, −2.36040609819935751347705098917, −0.36985154384118916324762814411,
0.36985154384118916324762814411, 2.36040609819935751347705098917, 3.10806833600470773382331709745, 3.61943960348352280076080732345, 4.42137501373968687167020578592, 5.12443958063506590635176336211, 5.73543405007254104994403528215, 6.64577003571098300904118715846, 7.24613236490352677519154150787, 7.73603498573659811230938579823