L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 2·15-s − 6·17-s − 7·19-s − 2·21-s + 23-s − 4·25-s + 4·27-s − 5·29-s − 7·31-s + 2·33-s − 35-s − 2·37-s + 2·39-s + 6·41-s + 43-s − 45-s + 7·47-s + 49-s + 12·51-s − 13·53-s + 55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 1.45·17-s − 1.60·19-s − 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.769·27-s − 0.928·29-s − 1.25·31-s + 0.348·33-s − 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.152·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s + 1.68·51-s − 1.78·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2454990878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2454990878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.55473416294089738858284930941, −7.24322299830921376252435579808, −6.16427793242467904808511223661, −5.97105697699043253490722348142, −4.95004599450949897085522612360, −4.47891540048586343781737092401, −3.75866605852823641029261154447, −2.51772331609807905380370376223, −1.73667745344387506660832475871, −0.25001082515270054092396656263,
0.25001082515270054092396656263, 1.73667745344387506660832475871, 2.51772331609807905380370376223, 3.75866605852823641029261154447, 4.47891540048586343781737092401, 4.95004599450949897085522612360, 5.97105697699043253490722348142, 6.16427793242467904808511223661, 7.24322299830921376252435579808, 7.55473416294089738858284930941