| L(s) = 1 | − 2.56·3-s − 2.44·5-s − 1.80·7-s + 3.58·9-s − 2.13·11-s + 4.47·13-s + 6.28·15-s − 0.551·17-s + 1.19·19-s + 4.64·21-s + 0.789·23-s + 0.994·25-s − 1.50·27-s − 4.78·29-s − 4.84·31-s + 5.48·33-s + 4.42·35-s − 0.0175·37-s − 11.4·39-s + 6.24·41-s − 2.48·43-s − 8.78·45-s − 13.2·47-s − 3.72·49-s + 1.41·51-s + 4.32·53-s + 5.23·55-s + ⋯ |
| L(s) = 1 | − 1.48·3-s − 1.09·5-s − 0.683·7-s + 1.19·9-s − 0.644·11-s + 1.24·13-s + 1.62·15-s − 0.133·17-s + 0.274·19-s + 1.01·21-s + 0.164·23-s + 0.198·25-s − 0.290·27-s − 0.889·29-s − 0.869·31-s + 0.954·33-s + 0.748·35-s − 0.00288·37-s − 1.83·39-s + 0.974·41-s − 0.378·43-s − 1.30·45-s − 1.93·47-s − 0.532·49-s + 0.198·51-s + 0.593·53-s + 0.705·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3063594482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3063594482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.551T + 17T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 23 | \( 1 - 0.789T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 + 4.84T + 31T^{2} \) |
| 37 | \( 1 + 0.0175T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 1.93T + 61T^{2} \) |
| 67 | \( 1 + 5.88T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 0.170T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−7.922925156575258959313037049043, −7.27458928439766135898793430167, −6.56730231736573446700234095721, −5.92273805382148445242483373970, −5.35068716309594169018353984755, −4.48824465014648544953460693520, −3.75972613898701892898222332098, −3.03824280367237469311098980836, −1.50030501463659238638669718340, −0.32450503562078242513419484490,
0.32450503562078242513419484490, 1.50030501463659238638669718340, 3.03824280367237469311098980836, 3.75972613898701892898222332098, 4.48824465014648544953460693520, 5.35068716309594169018353984755, 5.92273805382148445242483373970, 6.56730231736573446700234095721, 7.27458928439766135898793430167, 7.922925156575258959313037049043
