| L(s) = 1 | − 2.44·5-s − 1.41·7-s − 3.46·11-s − 4.89·13-s + 4·17-s − 6.92·19-s − 5.65·23-s + 0.999·25-s − 2.44·29-s + 1.41·31-s + 3.46·35-s − 4.89·37-s + 4·41-s + 6.92·43-s + 5.65·47-s − 5·49-s + 7.34·53-s + 8.48·55-s + 13.8·59-s − 4.89·61-s + 11.9·65-s − 11.3·71-s − 4·73-s + 4.89·77-s − 7.07·79-s − 10.3·83-s − 9.79·85-s + ⋯ |
| L(s) = 1 | − 1.09·5-s − 0.534·7-s − 1.04·11-s − 1.35·13-s + 0.970·17-s − 1.58·19-s − 1.17·23-s + 0.199·25-s − 0.454·29-s + 0.254·31-s + 0.585·35-s − 0.805·37-s + 0.624·41-s + 1.05·43-s + 0.825·47-s − 0.714·49-s + 1.00·53-s + 1.14·55-s + 1.80·59-s − 0.627·61-s + 1.48·65-s − 1.34·71-s − 0.468·73-s + 0.558·77-s − 0.795·79-s − 1.14·83-s − 1.06·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4722981658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4722981658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 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
−8.179241106968710109652229542484, −7.56968261334377706744850402481, −7.14664586315026700853661347026, −6.07701350004751985649604477034, −5.39964650025734038334357711465, −4.43220427854209321098822670179, −3.88075901163517833051891350768, −2.88199546820798744855326852443, −2.13011582929890769845634910224, −0.35521274087390964463913027938,
0.35521274087390964463913027938, 2.13011582929890769845634910224, 2.88199546820798744855326852443, 3.88075901163517833051891350768, 4.43220427854209321098822670179, 5.39964650025734038334357711465, 6.07701350004751985649604477034, 7.14664586315026700853661347026, 7.56968261334377706744850402481, 8.179241106968710109652229542484
