| L(s) = 1 | − 2.55·3-s + 15.7·5-s − 25.1·7-s − 20.4·9-s − 15.8·11-s − 15.4·13-s − 40.2·15-s + 56.7·17-s + 107.·19-s + 64.4·21-s − 23·23-s + 122.·25-s + 121.·27-s + 267.·29-s + 35.0·31-s + 40.5·33-s − 396.·35-s − 84.9·37-s + 39.6·39-s + 296.·41-s − 353.·43-s − 322.·45-s − 86.6·47-s + 291.·49-s − 145.·51-s + 126.·53-s − 249.·55-s + ⋯ |
| L(s) = 1 | − 0.492·3-s + 1.40·5-s − 1.35·7-s − 0.757·9-s − 0.434·11-s − 0.330·13-s − 0.693·15-s + 0.810·17-s + 1.29·19-s + 0.669·21-s − 0.208·23-s + 0.982·25-s + 0.865·27-s + 1.71·29-s + 0.203·31-s + 0.213·33-s − 1.91·35-s − 0.377·37-s + 0.162·39-s + 1.13·41-s − 1.25·43-s − 1.06·45-s − 0.268·47-s + 0.849·49-s − 0.398·51-s + 0.328·53-s − 0.611·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.55T + 27T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 + 25.1T + 343T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 35.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 86.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 126.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 603.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 766.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.000991389657237037138956752145, −7.915817742651554666104812337060, −6.83544723988830596699057426821, −6.10930266451819008948164600170, −5.64394925923888431050723551809, −4.82555452758901574163683804352, −3.15555996760764495784993465668, −2.72668353865807251308029044285, −1.25033350857530458615822580454, 0,
1.25033350857530458615822580454, 2.72668353865807251308029044285, 3.15555996760764495784993465668, 4.82555452758901574163683804352, 5.64394925923888431050723551809, 6.10930266451819008948164600170, 6.83544723988830596699057426821, 7.915817742651554666104812337060, 9.000991389657237037138956752145
