| L(s) = 1 | − 5·5-s − 24.3·7-s − 26.8·11-s − 68.2·13-s − 61.8·17-s − 75.1·19-s − 43.3·23-s + 25·25-s + 174.·29-s + 222.·31-s + 121.·35-s + 67.1·37-s − 22.2·41-s − 84.7·43-s − 585.·47-s + 251.·49-s + 38.3·53-s + 134.·55-s + 92·59-s − 226.·61-s + 341.·65-s + 858.·67-s + 116.·71-s + 911.·73-s + 655.·77-s − 285.·79-s − 999.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.31·7-s − 0.737·11-s − 1.45·13-s − 0.882·17-s − 0.907·19-s − 0.393·23-s + 0.200·25-s + 1.12·29-s + 1.29·31-s + 0.588·35-s + 0.298·37-s − 0.0845·41-s − 0.300·43-s − 1.81·47-s + 0.734·49-s + 0.0994·53-s + 0.329·55-s + 0.203·59-s − 0.474·61-s + 0.651·65-s + 1.56·67-s + 0.195·71-s + 1.46·73-s + 0.970·77-s − 0.406·79-s − 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5458889240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5458889240\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| good | 7 | \( 1 + 24.3T + 343T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 38.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92T + 2.05e5T^{2} \) |
| 61 | \( 1 + 226.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 116.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 911.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 285.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 999.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 374.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
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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
−9.766009439412724389918061979468, −8.602615539461453790908363727526, −7.895331629371019626926421929848, −6.82170614364800102278932853488, −6.38288981780421370951668714170, −5.06387402262939474356702306310, −4.28668730939271111942249872291, −3.06606021409085581150303377446, −2.32853990848964681019277700029, −0.35892793162557326002364945228,
0.35892793162557326002364945228, 2.32853990848964681019277700029, 3.06606021409085581150303377446, 4.28668730939271111942249872291, 5.06387402262939474356702306310, 6.38288981780421370951668714170, 6.82170614364800102278932853488, 7.895331629371019626926421929848, 8.602615539461453790908363727526, 9.766009439412724389918061979468
