| L(s) = 1 | + 7·2-s + 9·3-s + 17·4-s − 25·5-s + 63·6-s + 12·7-s − 105·8-s + 81·9-s − 175·10-s + 112·11-s + 153·12-s − 974·13-s + 84·14-s − 225·15-s − 1.27e3·16-s + 2.18e3·17-s + 567·18-s + 1.42e3·19-s − 425·20-s + 108·21-s + 784·22-s + 3.21e3·23-s − 945·24-s + 625·25-s − 6.81e3·26-s + 729·27-s + 204·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.577·3-s + 0.531·4-s − 0.447·5-s + 0.714·6-s + 0.0925·7-s − 0.580·8-s + 1/3·9-s − 0.553·10-s + 0.279·11-s + 0.306·12-s − 1.59·13-s + 0.114·14-s − 0.258·15-s − 1.24·16-s + 1.83·17-s + 0.412·18-s + 0.902·19-s − 0.237·20-s + 0.0534·21-s + 0.345·22-s + 1.26·23-s − 0.334·24-s + 1/5·25-s − 1.97·26-s + 0.192·27-s + 0.0491·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.314690733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.314690733\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 7 T + p^{5} T^{2} \) |
| 7 | \( 1 - 12 T + p^{5} T^{2} \) |
| 11 | \( 1 - 112 T + p^{5} T^{2} \) |
| 13 | \( 1 + 974 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2182 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1420 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4150 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5688 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6482 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5402 T + p^{5} T^{2} \) |
| 43 | \( 1 + 21764 T + p^{5} T^{2} \) |
| 47 | \( 1 + 368 T + p^{5} T^{2} \) |
| 53 | \( 1 - 12586 T + p^{5} T^{2} \) |
| 59 | \( 1 + 25520 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11782 T + p^{5} T^{2} \) |
| 67 | \( 1 + 13188 T + p^{5} T^{2} \) |
| 71 | \( 1 + 35968 T + p^{5} T^{2} \) |
| 73 | \( 1 - 73186 T + p^{5} T^{2} \) |
| 79 | \( 1 + 52440 T + p^{5} T^{2} \) |
| 83 | \( 1 - 69036 T + p^{5} T^{2} \) |
| 89 | \( 1 + 33870 T + p^{5} T^{2} \) |
| 97 | \( 1 - 143042 T + p^{5} T^{2} \) |
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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
−18.56806804791753158209280315134, −16.67711229778738368650618285473, −14.96423166989988306238358894425, −14.40959883142525509487712137561, −12.88689811845441631635567126720, −11.80327658753742442307100363017, −9.513779739225472188745677407481, −7.45479289603636973054992674117, −5.11509813201056562896930058844, −3.28780360799114787730035245347,
3.28780360799114787730035245347, 5.11509813201056562896930058844, 7.45479289603636973054992674117, 9.513779739225472188745677407481, 11.80327658753742442307100363017, 12.88689811845441631635567126720, 14.40959883142525509487712137561, 14.96423166989988306238358894425, 16.67711229778738368650618285473, 18.56806804791753158209280315134
