| L(s) = 1 | + 6.42·3-s − 14.1·5-s + 14.0·7-s + 14.2·9-s − 55.5·11-s + 18.3·13-s − 90.5·15-s + 10.0·17-s + 161.·19-s + 89.9·21-s + 23·23-s + 73.8·25-s − 81.9·27-s − 183.·29-s + 144.·31-s − 356.·33-s − 197.·35-s − 181.·37-s + 117.·39-s + 77.7·41-s + 315.·43-s − 200.·45-s + 524.·47-s − 146.·49-s + 64.3·51-s − 73.7·53-s + 782.·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 1.26·5-s + 0.756·7-s + 0.527·9-s − 1.52·11-s + 0.390·13-s − 1.55·15-s + 0.143·17-s + 1.94·19-s + 0.934·21-s + 0.208·23-s + 0.591·25-s − 0.584·27-s − 1.17·29-s + 0.838·31-s − 1.88·33-s − 0.954·35-s − 0.806·37-s + 0.482·39-s + 0.296·41-s + 1.11·43-s − 0.665·45-s + 1.62·47-s − 0.427·49-s + 0.176·51-s − 0.191·53-s + 1.91·55-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
| \(L(2)\) |
\(\approx\) |
\(2.599425481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599425481\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 6.42T + 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 - 14.0T + 343T^{2} \) |
| 11 | \( 1 + 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 161.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 524.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 73.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 236.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 806.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 71.2T + 9.12e5T^{2} \) |
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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
−8.950855599949069538098776724765, −8.091431692991254144958196760889, −7.76150685646214158097807197095, −7.29079861910023182420245080960, −5.63334131587720319602008650971, −4.85927837335723460659523530376, −3.75917336624620714749656565791, −3.14500652217727997119470192244, −2.18046875430725992814412923950, −0.73775011408151792582967128454,
0.73775011408151792582967128454, 2.18046875430725992814412923950, 3.14500652217727997119470192244, 3.75917336624620714749656565791, 4.85927837335723460659523530376, 5.63334131587720319602008650971, 7.29079861910023182420245080960, 7.76150685646214158097807197095, 8.091431692991254144958196760889, 8.950855599949069538098776724765
