| L(s) = 1 | − 1.56·3-s + 7-s − 0.561·9-s − 6.12·11-s + 2·13-s + 1.56·17-s + 3.56·19-s − 1.56·21-s + 1.43·23-s + 5.56·27-s + 3.43·29-s − 9.12·31-s + 9.56·33-s + 8.80·37-s − 3.12·39-s − 2.43·41-s + 6.56·43-s − 8.24·47-s + 49-s − 2.43·51-s − 1.12·53-s − 5.56·57-s + 11.3·59-s + 11.1·61-s − 0.561·63-s + 7.87·67-s − 2.24·69-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.377·7-s − 0.187·9-s − 1.84·11-s + 0.554·13-s + 0.378·17-s + 0.817·19-s − 0.340·21-s + 0.299·23-s + 1.07·27-s + 0.638·29-s − 1.63·31-s + 1.66·33-s + 1.44·37-s − 0.500·39-s − 0.380·41-s + 1.00·43-s − 1.20·47-s + 0.142·49-s − 0.341·51-s − 0.154·53-s − 0.736·57-s + 1.48·59-s + 1.42·61-s − 0.0707·63-s + 0.962·67-s − 0.270·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.011834037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.011834037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 6.43T + 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 + 1.31T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.764886541843858359071372728501, −8.623794184886935393997146175628, −7.928524300526593613706540030669, −7.16615429425786812669629194646, −6.03755799412375172387116788170, −5.38190326402091152015642817625, −4.85377048861373154030166071134, −3.44642196103307571227694096033, −2.37833347873945477786681427203, −0.74789096578835579845080643150,
0.74789096578835579845080643150, 2.37833347873945477786681427203, 3.44642196103307571227694096033, 4.85377048861373154030166071134, 5.38190326402091152015642817625, 6.03755799412375172387116788170, 7.16615429425786812669629194646, 7.928524300526593613706540030669, 8.623794184886935393997146175628, 9.764886541843858359071372728501
