| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s − 4.17·11-s − 5.03·13-s − 48.0·17-s − 22.5·19-s − 21·21-s + 125.·23-s − 27·27-s + 81.2·29-s − 29.0·31-s + 12.5·33-s − 9.69·37-s + 15.1·39-s − 259.·41-s + 138.·43-s − 475.·47-s + 49·49-s + 144.·51-s + 654.·53-s + 67.5·57-s − 343.·59-s − 215.·61-s + 63·63-s − 829.·67-s − 377.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s − 0.114·11-s − 0.107·13-s − 0.685·17-s − 0.271·19-s − 0.218·21-s + 1.13·23-s − 0.192·27-s + 0.520·29-s − 0.168·31-s + 0.0659·33-s − 0.0430·37-s + 0.0620·39-s − 0.989·41-s + 0.490·43-s − 1.47·47-s + 0.142·49-s + 0.395·51-s + 1.69·53-s + 0.157·57-s − 0.758·59-s − 0.452·61-s + 0.125·63-s − 1.51·67-s − 0.657·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 + 4.17T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.03T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 22.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 81.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.69T + 5.06e4T^{2} \) |
| 41 | \( 1 + 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 475.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 654.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 215.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 829.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 165.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 134.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 459.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 594.T + 9.12e5T^{2} \) |
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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
−8.435970896220072960333085460985, −7.46540600344029153647149382570, −6.78695764310600598252522380887, −6.00783926154086530687324442034, −5.05104887374483229983131576078, −4.52280829835768138373401645980, −3.38910968636067572124444604330, −2.26380918092639339739532642645, −1.17350015446543188972321881684, 0,
1.17350015446543188972321881684, 2.26380918092639339739532642645, 3.38910968636067572124444604330, 4.52280829835768138373401645980, 5.05104887374483229983131576078, 6.00783926154086530687324442034, 6.78695764310600598252522380887, 7.46540600344029153647149382570, 8.435970896220072960333085460985
