| L(s) = 1 | − 3·3-s − 5·5-s − 7·7-s + 9·9-s − 37.4·11-s + 29.0·13-s + 15·15-s + 58.4·17-s + 54.5·19-s + 21·21-s − 161.·23-s + 25·25-s − 27·27-s + 137.·29-s − 154.·31-s + 112.·33-s + 35·35-s − 350.·37-s − 87.0·39-s + 353.·41-s + 518.·43-s − 45·45-s + 542.·47-s + 49·49-s − 175.·51-s + 305.·53-s + 187.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 1.02·11-s + 0.619·13-s + 0.258·15-s + 0.833·17-s + 0.659·19-s + 0.218·21-s − 1.46·23-s + 0.200·25-s − 0.192·27-s + 0.880·29-s − 0.896·31-s + 0.591·33-s + 0.169·35-s − 1.55·37-s − 0.357·39-s + 1.34·41-s + 1.83·43-s − 0.149·45-s + 1.68·47-s + 0.142·49-s − 0.481·51-s + 0.791·53-s + 0.458·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−8.531756647116448101446720585050, −7.65799538734073440975979290978, −7.15152862733662241905964892920, −5.88166212268994993932282941827, −5.58669225128133447694363234460, −4.38301460074768451552671671976, −3.56974716056325902453382840958, −2.49988520297808149484077279092, −1.07404457693876035493318885050, 0,
1.07404457693876035493318885050, 2.49988520297808149484077279092, 3.56974716056325902453382840958, 4.38301460074768451552671671976, 5.58669225128133447694363234460, 5.88166212268994993932282941827, 7.15152862733662241905964892920, 7.65799538734073440975979290978, 8.531756647116448101446720585050
