| L(s) = 1 | − 2-s + 0.618·3-s + 4-s − 0.618·6-s − 8-s − 0.618·9-s + 0.618·12-s + 16-s + 1.61·17-s + 0.618·18-s + 1.61·19-s − 0.618·24-s + 25-s − 27-s − 32-s − 1.61·34-s − 0.618·36-s − 1.61·38-s − 0.618·41-s − 0.618·43-s + 0.618·48-s + 49-s − 50-s + 1.00·51-s + 54-s + 1.00·57-s − 1.61·59-s + ⋯ |
| L(s) = 1 | − 2-s + 0.618·3-s + 4-s − 0.618·6-s − 8-s − 0.618·9-s + 0.618·12-s + 16-s + 1.61·17-s + 0.618·18-s + 1.61·19-s − 0.618·24-s + 25-s − 27-s − 32-s − 1.61·34-s − 0.618·36-s − 1.61·38-s − 0.618·41-s − 0.618·43-s + 0.618·48-s + 49-s − 50-s + 1.00·51-s + 54-s + 1.00·57-s − 1.61·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8189568752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8189568752\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.61T + T^{2} \) |
| 19 | \( 1 - 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.618T + T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.618T + T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−10.01647192132624934634680632287, −9.322341086282920957263983916231, −8.620026202525772816254659043348, −7.79491959774243289349524856575, −7.25525343873670708925087898445, −6.03321554057795925551637804577, −5.22500461694516380660787897559, −3.40797988988693528798181591233, −2.83607104763245697038228042632, −1.33221425363398548150472580559,
1.33221425363398548150472580559, 2.83607104763245697038228042632, 3.40797988988693528798181591233, 5.22500461694516380660787897559, 6.03321554057795925551637804577, 7.25525343873670708925087898445, 7.79491959774243289349524856575, 8.620026202525772816254659043348, 9.322341086282920957263983916231, 10.01647192132624934634680632287
