L(s) = 1 | − 2-s − 3-s + 4-s − 3.56·5-s + 6-s − 7-s − 8-s + 9-s + 3.56·10-s + 5.56·11-s − 12-s + 14-s + 3.56·15-s + 16-s + 6.68·17-s − 18-s + 1.56·19-s − 3.56·20-s + 21-s − 5.56·22-s + 6.68·23-s + 24-s + 7.68·25-s − 27-s − 28-s + 1.56·29-s − 3.56·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.59·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s + 1.67·11-s − 0.288·12-s + 0.267·14-s + 0.919·15-s + 0.250·16-s + 1.62·17-s − 0.235·18-s + 0.358·19-s − 0.796·20-s + 0.218·21-s − 1.18·22-s + 1.39·23-s + 0.204·24-s + 1.53·25-s − 0.192·27-s − 0.188·28-s + 0.289·29-s − 0.650·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.078533525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078533525\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.950582076466940698145219677834, −7.24833967352261023685898319759, −6.73188132240621946240011159912, −6.08346639879335781299519797261, −5.07760720067687761371554941761, −4.22998135057417449139168524993, −3.56513794513188689145523150626, −2.89827504331595590276262707192, −1.18989435134740020733538435282, −0.75371095167969600174767616774,
0.75371095167969600174767616774, 1.18989435134740020733538435282, 2.89827504331595590276262707192, 3.56513794513188689145523150626, 4.22998135057417449139168524993, 5.07760720067687761371554941761, 6.08346639879335781299519797261, 6.73188132240621946240011159912, 7.24833967352261023685898319759, 7.950582076466940698145219677834