| L(s) = 1 | + 4·2-s − 3·3-s + 8·4-s − 4·5-s − 12·6-s − 7·7-s + 9·9-s − 16·10-s + 62·11-s − 24·12-s − 62·13-s − 28·14-s + 12·15-s − 64·16-s + 84·17-s + 36·18-s + 100·19-s − 32·20-s + 21·21-s + 248·22-s − 42·23-s − 109·25-s − 248·26-s − 27·27-s − 56·28-s − 10·29-s + 48·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.357·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.505·10-s + 1.69·11-s − 0.577·12-s − 1.32·13-s − 0.534·14-s + 0.206·15-s − 16-s + 1.19·17-s + 0.471·18-s + 1.20·19-s − 0.357·20-s + 0.218·21-s + 2.40·22-s − 0.380·23-s − 0.871·25-s − 1.87·26-s − 0.192·27-s − 0.377·28-s − 0.0640·29-s + 0.292·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.652586624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.652586624\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 62 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 10 T + p^{3} T^{2} \) |
| 31 | \( 1 + 48 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 - 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 678 T + p^{3} T^{2} \) |
| 73 | \( 1 + 642 T + p^{3} T^{2} \) |
| 79 | \( 1 - 740 T + p^{3} T^{2} \) |
| 83 | \( 1 - 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 200 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1266 T + p^{3} T^{2} \) |
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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
−17.40127996782650162168698349005, −16.20368624387748991924412491439, −14.85183094628299495439960961276, −13.88852072583434235810271714496, −12.18734969592006225633160749429, −11.84300799973327321430232327034, −9.655720314736560965383428982413, −7.01970943968790635781203522162, −5.46943372320388946773529265287, −3.78050195514546319844079133815,
3.78050195514546319844079133815, 5.46943372320388946773529265287, 7.01970943968790635781203522162, 9.655720314736560965383428982413, 11.84300799973327321430232327034, 12.18734969592006225633160749429, 13.88852072583434235810271714496, 14.85183094628299495439960961276, 16.20368624387748991924412491439, 17.40127996782650162168698349005
