L(s) = 1 | − 3·2-s − 3·3-s + 4-s − 5·5-s + 9·6-s − 20·7-s + 21·8-s + 9·9-s + 15·10-s − 3·12-s − 74·13-s + 60·14-s + 15·15-s − 71·16-s − 54·17-s − 27·18-s + 124·19-s − 5·20-s + 60·21-s − 120·23-s − 63·24-s + 25·25-s + 222·26-s − 27·27-s − 20·28-s + 78·29-s − 45·30-s + ⋯ |
L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s − 0.447·5-s + 0.612·6-s − 1.07·7-s + 0.928·8-s + 1/3·9-s + 0.474·10-s − 0.0721·12-s − 1.57·13-s + 1.14·14-s + 0.258·15-s − 1.10·16-s − 0.770·17-s − 0.353·18-s + 1.49·19-s − 0.0559·20-s + 0.623·21-s − 1.08·23-s − 0.535·24-s + 1/5·25-s + 1.67·26-s − 0.192·27-s − 0.134·28-s + 0.499·29-s − 0.273·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.596074352717307228140404880788, −7.70397283425504460239390282451, −7.10533707220602567085769972749, −6.40051164897602578373357097884, −5.18164795671430455818158496530, −4.50612464656898946247432458992, −3.37516262709443217785228429144, −2.17883285999070286816097996558, −0.74667345425623936623707578537, 0,
0.74667345425623936623707578537, 2.17883285999070286816097996558, 3.37516262709443217785228429144, 4.50612464656898946247432458992, 5.18164795671430455818158496530, 6.40051164897602578373357097884, 7.10533707220602567085769972749, 7.70397283425504460239390282451, 8.596074352717307228140404880788