| L(s) = 1 | − 0.192·3-s − 5·5-s + 11.9·7-s − 26.9·9-s + 51.6·11-s − 44.7·13-s + 0.964·15-s − 39.0·17-s − 4.72·19-s − 2.30·21-s + 23·23-s + 25·25-s + 10.4·27-s + 292.·29-s + 11.1·31-s − 9.96·33-s − 59.6·35-s − 257.·37-s + 8.63·39-s + 413.·41-s − 364.·43-s + 134.·45-s − 187.·47-s − 200.·49-s + 7.52·51-s + 11.2·53-s − 258.·55-s + ⋯ |
| L(s) = 1 | − 0.0371·3-s − 0.447·5-s + 0.644·7-s − 0.998·9-s + 1.41·11-s − 0.955·13-s + 0.0165·15-s − 0.556·17-s − 0.0570·19-s − 0.0239·21-s + 0.208·23-s + 0.200·25-s + 0.0741·27-s + 1.87·29-s + 0.0643·31-s − 0.0525·33-s − 0.288·35-s − 1.14·37-s + 0.0354·39-s + 1.57·41-s − 1.29·43-s + 0.446·45-s − 0.581·47-s − 0.584·49-s + 0.0206·51-s + 0.0290·53-s − 0.633·55-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 | \( 1 + 0.192T + 27T^{2} \) |
| 7 | \( 1 - 11.9T + 343T^{2} \) |
| 11 | \( 1 - 51.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.72T + 6.85e3T^{2} \) |
| 29 | \( 1 - 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 413.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 11.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 362.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 584.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 742.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 401.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 293.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 154.T + 9.12e5T^{2} \) |
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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
−8.564315496904435486082071382345, −7.80430757801260616352040715099, −6.84497754791203054292895008881, −6.23788077913632105259022264709, −5.06150026976511776084413453603, −4.48312896583831039730669186312, −3.42330541293075433978201348740, −2.43914518301184209029814190538, −1.23839204646157960764267333940, 0,
1.23839204646157960764267333940, 2.43914518301184209029814190538, 3.42330541293075433978201348740, 4.48312896583831039730669186312, 5.06150026976511776084413453603, 6.23788077913632105259022264709, 6.84497754791203054292895008881, 7.80430757801260616352040715099, 8.564315496904435486082071382345
