| L(s) = 1 | + 2·2-s + 3-s + 4·4-s − 5·5-s + 2·6-s + 8·8-s − 26·9-s − 10·10-s − 30·11-s + 4·12-s + 44·13-s − 5·15-s + 16·16-s − 24·17-s − 52·18-s + 2·19-s − 20·20-s − 60·22-s − 183·23-s + 8·24-s + 25·25-s + 88·26-s − 53·27-s − 279·29-s − 10·30-s − 40·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s + 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.822·11-s + 0.0962·12-s + 0.938·13-s − 0.0860·15-s + 1/4·16-s − 0.342·17-s − 0.680·18-s + 0.0241·19-s − 0.223·20-s − 0.581·22-s − 1.65·23-s + 0.0680·24-s + 1/5·25-s + 0.663·26-s − 0.377·27-s − 1.78·29-s − 0.0608·30-s − 0.231·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 279 T + p^{3} T^{2} \) |
| 31 | \( 1 + 40 T + p^{3} T^{2} \) |
| 37 | \( 1 + 76 T + p^{3} T^{2} \) |
| 41 | \( 1 + 423 T + p^{3} T^{2} \) |
| 43 | \( 1 - 305 T + p^{3} T^{2} \) |
| 47 | \( 1 - 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 462 T + p^{3} T^{2} \) |
| 61 | \( 1 - 281 T + p^{3} T^{2} \) |
| 67 | \( 1 + 499 T + p^{3} T^{2} \) |
| 71 | \( 1 + 534 T + p^{3} T^{2} \) |
| 73 | \( 1 - 800 T + p^{3} T^{2} \) |
| 79 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 597 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−10.38166331653777360446089994101, −9.063009495112211704355361125741, −8.186613377546532040796367724073, −7.41342166951706331423825958947, −6.10715508185741108634646563156, −5.44743905038438995143515406801, −4.13056484692384285928622488378, −3.25318206359597413555533974387, −2.02790619042893604715846005225, 0,
2.02790619042893604715846005225, 3.25318206359597413555533974387, 4.13056484692384285928622488378, 5.44743905038438995143515406801, 6.10715508185741108634646563156, 7.41342166951706331423825958947, 8.186613377546532040796367724073, 9.063009495112211704355361125741, 10.38166331653777360446089994101
