| L(s) = 1 | + 2-s + 2.41·3-s + 4-s − 5-s + 2.41·6-s + 8-s + 2.82·9-s − 10-s + 3.41·11-s + 2.41·12-s − 2.24·13-s − 2.41·15-s + 16-s + 7.82·17-s + 2.82·18-s + 2.82·19-s − 20-s + 3.41·22-s − 4·23-s + 2.41·24-s − 4·25-s − 2.24·26-s − 0.414·27-s − 1.82·29-s − 2.41·30-s + 5.58·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s − 0.447·5-s + 0.985·6-s + 0.353·8-s + 0.942·9-s − 0.316·10-s + 1.02·11-s + 0.696·12-s − 0.621·13-s − 0.623·15-s + 0.250·16-s + 1.89·17-s + 0.666·18-s + 0.648·19-s − 0.223·20-s + 0.727·22-s − 0.834·23-s + 0.492·24-s − 0.800·25-s − 0.439·26-s − 0.0797·27-s − 0.339·29-s − 0.440·30-s + 1.00·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.133179757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.133179757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 0.828T + 59T^{2} \) |
| 61 | \( 1 + 8.65T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 2.17T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.224450430043043744434337874556, −7.68778270235596672387908357711, −7.28050082705256962003315768450, −6.15093491391207252972665500173, −5.46044346161504072681157314901, −4.31372986307830358900138359113, −3.76839528609772804238067822461, −3.12110792398171627437740909344, −2.29086931107915874822442724970, −1.19514282988006579913876045042,
1.19514282988006579913876045042, 2.29086931107915874822442724970, 3.12110792398171627437740909344, 3.76839528609772804238067822461, 4.31372986307830358900138359113, 5.46044346161504072681157314901, 6.15093491391207252972665500173, 7.28050082705256962003315768450, 7.68778270235596672387908357711, 8.224450430043043744434337874556
