| L(s) = 1 | − 5.15·2-s + 18.5·4-s + 5·5-s − 54.1·8-s − 25.7·10-s + 14.0·11-s + 10.5·13-s + 130.·16-s + 131.·17-s − 38.6·19-s + 92.6·20-s − 72.1·22-s + 102.·23-s + 25·25-s − 54.5·26-s + 232.·29-s + 165.·31-s − 240.·32-s − 678.·34-s + 280.·37-s + 199.·38-s − 270.·40-s − 122.·41-s − 431.·43-s + 259.·44-s − 527.·46-s + 295.·47-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.31·4-s + 0.447·5-s − 2.39·8-s − 0.814·10-s + 0.383·11-s + 0.225·13-s + 2.04·16-s + 1.88·17-s − 0.467·19-s + 1.03·20-s − 0.699·22-s + 0.929·23-s + 0.200·25-s − 0.411·26-s + 1.48·29-s + 0.958·31-s − 1.32·32-s − 3.42·34-s + 1.24·37-s + 0.850·38-s − 1.07·40-s − 0.465·41-s − 1.53·43-s + 0.888·44-s − 1.69·46-s + 0.917·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.308605382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308605382\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 11 | \( 1 - 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 431.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 295.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 566.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 188.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 176.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 190.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 518.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
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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
−8.683819594546744810699216992940, −8.155876682119363905892582279470, −7.38250975439020469939141615089, −6.57063603692609423944869860351, −5.98443781652427025225809367700, −4.84531799799000854144663536162, −3.34678956585723338180085909200, −2.50800385796252841676671975638, −1.34349783795088202659771518824, −0.77997796519220282639750815185,
0.77997796519220282639750815185, 1.34349783795088202659771518824, 2.50800385796252841676671975638, 3.34678956585723338180085909200, 4.84531799799000854144663536162, 5.98443781652427025225809367700, 6.57063603692609423944869860351, 7.38250975439020469939141615089, 8.155876682119363905892582279470, 8.683819594546744810699216992940
