| L(s) = 1 | + 5.04·3-s + 5·5-s − 5.03·7-s − 1.58·9-s + 5.58·11-s + 62.7·13-s + 25.2·15-s − 19.7·17-s − 158.·19-s − 25.3·21-s + 23·23-s + 25·25-s − 144.·27-s − 35.5·29-s − 282.·31-s + 28.1·33-s − 25.1·35-s − 139.·37-s + 316.·39-s + 227.·41-s − 436.·43-s − 7.94·45-s − 90.2·47-s − 317.·49-s − 99.5·51-s + 330.·53-s + 27.9·55-s + ⋯ |
| L(s) = 1 | + 0.970·3-s + 0.447·5-s − 0.271·7-s − 0.0588·9-s + 0.153·11-s + 1.33·13-s + 0.433·15-s − 0.281·17-s − 1.91·19-s − 0.263·21-s + 0.208·23-s + 0.200·25-s − 1.02·27-s − 0.227·29-s − 1.63·31-s + 0.148·33-s − 0.121·35-s − 0.621·37-s + 1.29·39-s + 0.867·41-s − 1.54·43-s − 0.0263·45-s − 0.280·47-s − 0.926·49-s − 0.273·51-s + 0.855·53-s + 0.0684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 5.04T + 27T^{2} \) |
| 7 | \( 1 + 5.03T + 343T^{2} \) |
| 11 | \( 1 - 5.58T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 158.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 35.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 90.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 330.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 568.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 85.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 310.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 158.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 106.T + 9.12e5T^{2} \) |
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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
−8.687538910526288491311533226670, −7.961370768054212673011522531561, −6.82830804644549820523945167846, −6.21170109989493008445531989644, −5.32674599544431729628031137942, −4.04355783712721860671411314220, −3.45395955870093679787812028194, −2.37205323841963248678629554614, −1.59742102849379825935529347848, 0,
1.59742102849379825935529347848, 2.37205323841963248678629554614, 3.45395955870093679787812028194, 4.04355783712721860671411314220, 5.32674599544431729628031137942, 6.21170109989493008445531989644, 6.82830804644549820523945167846, 7.961370768054212673011522531561, 8.687538910526288491311533226670
