| L(s) = 1 | + 5-s − 5.12·7-s − 11-s − 3.12·13-s − 3.12·17-s + 4·23-s + 25-s − 2·29-s − 5.12·35-s + 6·37-s − 6·41-s + 5.12·43-s − 4·47-s + 19.2·49-s + 8.24·53-s − 55-s − 4·59-s + 10·61-s − 3.12·65-s − 6.24·67-s + 6.24·71-s + 4.87·73-s + 5.12·77-s + 2.24·79-s − 11.3·83-s − 3.12·85-s + 16.2·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.93·7-s − 0.301·11-s − 0.866·13-s − 0.757·17-s + 0.834·23-s + 0.200·25-s − 0.371·29-s − 0.865·35-s + 0.986·37-s − 0.937·41-s + 0.781·43-s − 0.583·47-s + 2.74·49-s + 1.13·53-s − 0.134·55-s − 0.520·59-s + 1.28·61-s − 0.387·65-s − 0.763·67-s + 0.741·71-s + 0.570·73-s + 0.583·77-s + 0.252·79-s − 1.24·83-s − 0.338·85-s + 1.72·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.103566316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.103566316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.681927373356142504717409803471, −7.52444327848299020739194379237, −6.90183151129377935470332399835, −6.33194746375446376072284713970, −5.59790009145838515672466575544, −4.73108754265234899709649356983, −3.71360024776516049312484128673, −2.89622172001196878056084332535, −2.22995185898318267104759345975, −0.56887589256857787452940651825,
0.56887589256857787452940651825, 2.22995185898318267104759345975, 2.89622172001196878056084332535, 3.71360024776516049312484128673, 4.73108754265234899709649356983, 5.59790009145838515672466575544, 6.33194746375446376072284713970, 6.90183151129377935470332399835, 7.52444327848299020739194379237, 8.681927373356142504717409803471
