| L(s) = 1 | − 2-s − 4·3-s − 7·4-s + 4·6-s + 32·7-s + 15·8-s − 11·9-s + 40·11-s + 28·12-s + 66·13-s − 32·14-s + 41·16-s − 130·17-s + 11·18-s − 88·19-s − 128·21-s − 40·22-s − 23·23-s − 60·24-s − 66·26-s + 152·27-s − 224·28-s − 130·29-s + 40·31-s − 161·32-s − 160·33-s + 130·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.769·3-s − 7/8·4-s + 0.272·6-s + 1.72·7-s + 0.662·8-s − 0.407·9-s + 1.09·11-s + 0.673·12-s + 1.40·13-s − 0.610·14-s + 0.640·16-s − 1.85·17-s + 0.144·18-s − 1.06·19-s − 1.33·21-s − 0.387·22-s − 0.208·23-s − 0.510·24-s − 0.497·26-s + 1.08·27-s − 1.51·28-s − 0.832·29-s + 0.231·31-s − 0.889·32-s − 0.844·33-s + 0.655·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.154703085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154703085\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 130 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 - 40 T + p^{3} T^{2} \) |
| 37 | \( 1 - 334 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 612 T + p^{3} T^{2} \) |
| 61 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 496 T + p^{3} T^{2} \) |
| 71 | \( 1 - 248 T + p^{3} T^{2} \) |
| 73 | \( 1 + 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 296 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1296 T + p^{3} T^{2} \) |
| 89 | \( 1 + 646 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1438 T + p^{3} T^{2} \) |
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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
−10.64375005843870329570497998167, −9.123436795649546937251646872903, −8.666336021915546446640712983563, −7.965312001938591552089061195337, −6.55424522802932924334601234284, −5.71184951556162242094316952371, −4.56772135772405199709593288274, −4.09422835581358279170265454940, −1.87156300031122856839983602095, −0.74999984751775011355307723363,
0.74999984751775011355307723363, 1.87156300031122856839983602095, 4.09422835581358279170265454940, 4.56772135772405199709593288274, 5.71184951556162242094316952371, 6.55424522802932924334601234284, 7.965312001938591552089061195337, 8.666336021915546446640712983563, 9.123436795649546937251646872903, 10.64375005843870329570497998167
