L(s) = 1 | − 2-s − 0.0623·3-s + 4-s + 1.97·5-s + 0.0623·6-s − 0.185·7-s − 8-s − 2.99·9-s − 1.97·10-s − 1.12·11-s − 0.0623·12-s − 4.71·13-s + 0.185·14-s − 0.123·15-s + 16-s + 5.95·17-s + 2.99·18-s − 5.93·19-s + 1.97·20-s + 0.0115·21-s + 1.12·22-s + 5.58·23-s + 0.0623·24-s − 1.10·25-s + 4.71·26-s + 0.373·27-s − 0.185·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0359·3-s + 0.5·4-s + 0.883·5-s + 0.0254·6-s − 0.0701·7-s − 0.353·8-s − 0.998·9-s − 0.624·10-s − 0.338·11-s − 0.0179·12-s − 1.30·13-s + 0.0496·14-s − 0.0317·15-s + 0.250·16-s + 1.44·17-s + 0.706·18-s − 1.36·19-s + 0.441·20-s + 0.00252·21-s + 0.239·22-s + 1.16·23-s + 0.0127·24-s − 0.220·25-s + 0.924·26-s + 0.0719·27-s − 0.0350·28-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
\(L(1)\) |
\(\approx\) |
\(1.173366640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.173366640\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0623T + 3T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 + 0.185T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 + 9.23T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
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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
−8.133719882751753170803868332702, −7.50980058084933410694223463119, −6.65327635670223187092200063158, −6.01017971446729134948164271271, −5.36127362048983821337568176089, −4.66804222408589103228299485902, −3.27724904705104646981626821645, −2.62339628095552761826606721735, −1.89350536153172085554481984881, −0.61178450423065832554913318428,
0.61178450423065832554913318428, 1.89350536153172085554481984881, 2.62339628095552761826606721735, 3.27724904705104646981626821645, 4.66804222408589103228299485902, 5.36127362048983821337568176089, 6.01017971446729134948164271271, 6.65327635670223187092200063158, 7.50980058084933410694223463119, 8.133719882751753170803868332702