| L(s) = 1 | − 4.27·3-s − 1.27·5-s − 12.0·7-s − 8.72·9-s + 18.9·11-s + 55.9·13-s + 5.45·15-s − 89.7·17-s − 19·19-s + 51.7·21-s + 135.·23-s − 123.·25-s + 152.·27-s + 102.·29-s + 103.·31-s − 80.9·33-s + 15.4·35-s − 29.6·37-s − 239.·39-s + 234.·41-s − 53.3·43-s + 11.1·45-s + 33.3·47-s − 196.·49-s + 383.·51-s − 93.1·53-s − 24.1·55-s + ⋯ |
| L(s) = 1 | − 0.822·3-s − 0.114·5-s − 0.653·7-s − 0.323·9-s + 0.518·11-s + 1.19·13-s + 0.0938·15-s − 1.28·17-s − 0.229·19-s + 0.537·21-s + 1.22·23-s − 0.986·25-s + 1.08·27-s + 0.656·29-s + 0.600·31-s − 0.426·33-s + 0.0744·35-s − 0.131·37-s − 0.981·39-s + 0.894·41-s − 0.189·43-s + 0.0368·45-s + 0.103·47-s − 0.573·49-s + 1.05·51-s − 0.241·53-s − 0.0591·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 4.27T + 27T^{2} \) |
| 5 | \( 1 + 1.27T + 125T^{2} \) |
| 7 | \( 1 + 12.0T + 343T^{2} \) |
| 11 | \( 1 - 18.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 29.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 53.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 93.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 637.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 125.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 119.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 18.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 394.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 303.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 394.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.32e3T + 9.12e5T^{2} \) |
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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
−8.887497458767636168413871775978, −8.334349528544635124455707922369, −6.96115110044977149974241134101, −6.40285147943909509277634607177, −5.75752419631031203929610974460, −4.66523560019041847585610278691, −3.74342898057647677627976120129, −2.64395079291579370652894924229, −1.12551167233205896775023929478, 0,
1.12551167233205896775023929478, 2.64395079291579370652894924229, 3.74342898057647677627976120129, 4.66523560019041847585610278691, 5.75752419631031203929610974460, 6.40285147943909509277634607177, 6.96115110044977149974241134101, 8.334349528544635124455707922369, 8.887497458767636168413871775978
