| L(s) = 1 | + 2-s + 2.19·3-s + 4-s + 2.79·5-s + 2.19·6-s + 2.81·7-s + 8-s + 1.81·9-s + 2.79·10-s + 0.925·11-s + 2.19·12-s + 2.00·13-s + 2.81·14-s + 6.14·15-s + 16-s + 3.44·17-s + 1.81·18-s − 2.93·19-s + 2.79·20-s + 6.18·21-s + 0.925·22-s − 6.85·23-s + 2.19·24-s + 2.83·25-s + 2.00·26-s − 2.59·27-s + 2.81·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 1.25·5-s + 0.896·6-s + 1.06·7-s + 0.353·8-s + 0.606·9-s + 0.885·10-s + 0.279·11-s + 0.633·12-s + 0.557·13-s + 0.752·14-s + 1.58·15-s + 0.250·16-s + 0.834·17-s + 0.428·18-s − 0.672·19-s + 0.625·20-s + 1.34·21-s + 0.197·22-s − 1.42·23-s + 0.448·24-s + 0.567·25-s + 0.393·26-s − 0.499·27-s + 0.532·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.481874266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.481874266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 0.925T + 11T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 + 7.71T + 29T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 9.33T + 61T^{2} \) |
| 67 | \( 1 - 2.84T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
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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
−8.054610417750830179772040662190, −7.52807029150114915075131546518, −6.56590216090658704305760886581, −5.68540008361099248414393886752, −5.43230773866293932006330993013, −4.16202268843620686235926113285, −3.76634020107113953156475016940, −2.65070463047840337778806525130, −2.01967242944679047942913556603, −1.46242088964907785414736456952,
1.46242088964907785414736456952, 2.01967242944679047942913556603, 2.65070463047840337778806525130, 3.76634020107113953156475016940, 4.16202268843620686235926113285, 5.43230773866293932006330993013, 5.68540008361099248414393886752, 6.56590216090658704305760886581, 7.52807029150114915075131546518, 8.054610417750830179772040662190
