L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 3·11-s − 5·13-s + 16-s + 3·17-s − 5·19-s − 3·20-s − 3·22-s + 3·23-s + 4·25-s + 5·26-s + 3·29-s + 4·31-s − 32-s − 3·34-s − 7·37-s + 5·38-s + 3·40-s − 9·41-s + 11·43-s + 3·44-s − 3·46-s − 4·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.38·13-s + 1/4·16-s + 0.727·17-s − 1.14·19-s − 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s + 0.980·26-s + 0.557·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.15·37-s + 0.811·38-s + 0.474·40-s − 1.40·41-s + 1.67·43-s + 0.452·44-s − 0.442·46-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.56021616894276436533430184910, −6.97532561715796265164935477501, −6.45550288821213053253271546443, −5.35676393503085625688669955717, −4.55753516559170706157541556434, −3.88319219649919042878449799717, −3.08749423716812866973429359244, −2.17957417844442011176890832745, −0.993697501923244120843566516159, 0,
0.993697501923244120843566516159, 2.17957417844442011176890832745, 3.08749423716812866973429359244, 3.88319219649919042878449799717, 4.55753516559170706157541556434, 5.35676393503085625688669955717, 6.45550288821213053253271546443, 6.97532561715796265164935477501, 7.56021616894276436533430184910