| L(s) = 1 | + 1.53·2-s + 0.347·3-s + 1.34·4-s + 0.532·6-s + 0.347·7-s + 0.532·8-s − 0.879·9-s + 0.467·12-s + 0.532·14-s − 0.532·16-s − 1.34·18-s + 1.53·19-s + 0.120·21-s − 1.87·23-s + 0.184·24-s + 25-s − 0.652·27-s + 0.467·28-s − 31-s − 1.34·32-s − 1.18·36-s − 37-s + 2.34·38-s − 41-s + 0.184·42-s + 1.53·43-s − 2.87·46-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.347·3-s + 1.34·4-s + 0.532·6-s + 0.347·7-s + 0.532·8-s − 0.879·9-s + 0.467·12-s + 0.532·14-s − 0.532·16-s − 1.34·18-s + 1.53·19-s + 0.120·21-s − 1.87·23-s + 0.184·24-s + 25-s − 0.652·27-s + 0.467·28-s − 31-s − 1.34·32-s − 1.18·36-s − 37-s + 2.34·38-s − 41-s + 0.184·42-s + 1.53·43-s − 2.87·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.321448068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321448068\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + T^{2} \) |
| 3 | \( 1 - 0.347T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 + 1.87T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - 1.53T + T^{2} \) |
| 47 | \( 1 + 1.87T + T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.347T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.38648592015504680933720403728, −9.327412167602295514008624400239, −8.449804341833319040196381809237, −7.53608525234851157760367570565, −6.54130360215974781894454824804, −5.57374668159045178790180631849, −5.07693721660489733461266005609, −3.86194159611652715370162800235, −3.16861717143772265007164138154, −2.07532114180518672860907282457,
2.07532114180518672860907282457, 3.16861717143772265007164138154, 3.86194159611652715370162800235, 5.07693721660489733461266005609, 5.57374668159045178790180631849, 6.54130360215974781894454824804, 7.53608525234851157760367570565, 8.449804341833319040196381809237, 9.327412167602295514008624400239, 10.38648592015504680933720403728
