L(s) = 1 | + 3-s − 2.82·5-s + 9-s + 2.82·11-s − 2.82·15-s + 2.82·17-s + 4·19-s − 8.48·23-s + 3.00·25-s + 27-s + 2·29-s + 2.82·33-s − 6·37-s + 8.48·41-s − 11.3·43-s − 2.82·45-s + 8·47-s + 2.82·51-s + 6·53-s − 8.00·55-s + 4·57-s + 12·59-s + 5.65·61-s + 5.65·67-s − 8.48·69-s − 2.82·71-s − 5.65·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.26·5-s + 0.333·9-s + 0.852·11-s − 0.730·15-s + 0.685·17-s + 0.917·19-s − 1.76·23-s + 0.600·25-s + 0.192·27-s + 0.371·29-s + 0.492·33-s − 0.986·37-s + 1.32·41-s − 1.72·43-s − 0.421·45-s + 1.16·47-s + 0.396·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.56·59-s + 0.724·61-s + 0.691·67-s − 1.02·69-s − 0.335·71-s − 0.662·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.903045221\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.903045221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.418011186444346369260911153760, −7.49938692915668028334352650544, −7.20466612681010170960161840559, −6.18276001427499235957862745686, −5.32001893782356592288567186917, −4.23699768785466131205613081818, −3.80889437637996031292392665729, −3.11343650719143624076934396206, −1.93521082680014549546127995982, −0.75234923484583762022169548639,
0.75234923484583762022169548639, 1.93521082680014549546127995982, 3.11343650719143624076934396206, 3.80889437637996031292392665729, 4.23699768785466131205613081818, 5.32001893782356592288567186917, 6.18276001427499235957862745686, 7.20466612681010170960161840559, 7.49938692915668028334352650544, 8.418011186444346369260911153760