| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4.47·13-s − 15-s + 4.47·17-s + 21-s + 2.47·23-s + 25-s − 27-s − 0.472·29-s − 2.47·31-s − 35-s − 0.472·37-s − 4.47·39-s − 6.94·41-s − 1.52·43-s + 45-s − 6.47·47-s + 49-s − 4.47·51-s + 2·53-s + 4·59-s + 3.52·61-s − 63-s + 4.47·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 1.24·13-s − 0.258·15-s + 1.08·17-s + 0.218·21-s + 0.515·23-s + 0.200·25-s − 0.192·27-s − 0.0876·29-s − 0.444·31-s − 0.169·35-s − 0.0776·37-s − 0.716·39-s − 1.08·41-s − 0.232·43-s + 0.149·45-s − 0.944·47-s + 0.142·49-s − 0.626·51-s + 0.274·53-s + 0.520·59-s + 0.451·61-s − 0.125·63-s + 0.554·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732157838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732157838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−8.628915591270526553050448529572, −7.902637376282843827630202859548, −6.91150692810128692480479417832, −6.37156518835485607389422716935, −5.59165405293377127918187860998, −5.03121919910791507865073994186, −3.83387771265834082801310775156, −3.19093587872518693354887967796, −1.85332758713389238361254001689, −0.840062295966529603603512325649,
0.840062295966529603603512325649, 1.85332758713389238361254001689, 3.19093587872518693354887967796, 3.83387771265834082801310775156, 5.03121919910791507865073994186, 5.59165405293377127918187860998, 6.37156518835485607389422716935, 6.91150692810128692480479417832, 7.902637376282843827630202859548, 8.628915591270526553050448529572
