L(s) = 1 | − 3-s − 5-s + 2.82·7-s + 9-s + 2.82·11-s + 13-s + 15-s − 3.65·17-s − 1.17·19-s − 2.82·21-s + 4·23-s + 25-s − 27-s − 7.65·29-s − 6.82·31-s − 2.82·33-s − 2.82·35-s − 2·37-s − 39-s − 2·41-s − 9.65·43-s − 45-s − 6.82·47-s + 1.00·49-s + 3.65·51-s − 2·53-s − 2.82·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.06·7-s + 0.333·9-s + 0.852·11-s + 0.277·13-s + 0.258·15-s − 0.886·17-s − 0.268·19-s − 0.617·21-s + 0.834·23-s + 0.200·25-s − 0.192·27-s − 1.42·29-s − 1.22·31-s − 0.492·33-s − 0.478·35-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 1.47·43-s − 0.149·45-s − 0.996·47-s + 0.142·49-s + 0.512·51-s − 0.274·53-s − 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.71878625959832598377336505726, −6.84775307139191924109424521523, −6.47182732253031108130050905720, −5.33793574754763320591074739203, −4.93845505461745544324141015095, −4.05693533066938727376903134210, −3.46121125390384002069196025271, −2.03406364065911879725986757058, −1.35705521936650401298490173657, 0,
1.35705521936650401298490173657, 2.03406364065911879725986757058, 3.46121125390384002069196025271, 4.05693533066938727376903134210, 4.93845505461745544324141015095, 5.33793574754763320591074739203, 6.47182732253031108130050905720, 6.84775307139191924109424521523, 7.71878625959832598377336505726