| L(s) = 1 | − 5.52·2-s + 22.4·4-s + 6.08·5-s + 20.2·7-s − 80.0·8-s − 33.5·10-s + 48.8·11-s − 111.·14-s + 262.·16-s + 37.7·17-s + 120.·19-s + 136.·20-s − 269.·22-s − 74.8·23-s − 88.0·25-s + 456.·28-s + 112.·29-s + 113.·31-s − 807.·32-s − 208.·34-s + 123.·35-s − 85.7·37-s − 667.·38-s − 486.·40-s + 133.·41-s − 319.·43-s + 1.09e3·44-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 2.81·4-s + 0.543·5-s + 1.09·7-s − 3.53·8-s − 1.06·10-s + 1.33·11-s − 2.13·14-s + 4.09·16-s + 0.538·17-s + 1.45·19-s + 1.52·20-s − 2.61·22-s − 0.678·23-s − 0.704·25-s + 3.07·28-s + 0.721·29-s + 0.655·31-s − 4.46·32-s − 1.05·34-s + 0.595·35-s − 0.381·37-s − 2.84·38-s − 1.92·40-s + 0.510·41-s − 1.13·43-s + 3.76·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.462243442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462243442\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.52T + 8T^{2} \) |
| 5 | \( 1 - 6.08T + 125T^{2} \) |
| 7 | \( 1 - 20.2T + 343T^{2} \) |
| 11 | \( 1 - 48.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 37.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 384.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 121.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 220.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 975.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 539.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 229.T + 9.12e5T^{2} \) |
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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
−9.132451854085220418604475667472, −8.375232905121305298534302681149, −7.76637037157964222072596063576, −6.97681101709340997525040747387, −6.17320569044351987462359720102, −5.30677914166178575661990968458, −3.70260093899052320186209797149, −2.41521056355307107582995769827, −1.49756727281669206867883448415, −0.887607156019504275303688209714,
0.887607156019504275303688209714, 1.49756727281669206867883448415, 2.41521056355307107582995769827, 3.70260093899052320186209797149, 5.30677914166178575661990968458, 6.17320569044351987462359720102, 6.97681101709340997525040747387, 7.76637037157964222072596063576, 8.375232905121305298534302681149, 9.132451854085220418604475667472
