L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 11-s − 12-s − 2·13-s − 16-s − 18-s − 2·19-s − 22-s + 23-s + 3·24-s − 5·25-s + 2·26-s + 27-s − 10·29-s − 4·31-s − 5·32-s + 33-s − 36-s + 2·37-s + 2·38-s − 2·39-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.213·22-s + 0.208·23-s + 0.612·24-s − 25-s + 0.392·26-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.883·32-s + 0.174·33-s − 1/6·36-s + 0.328·37-s + 0.324·38-s − 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37191 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.97061764078193, −14.58949201386420, −14.23812148419434, −13.41691052383780, −13.22324117788800, −12.62942195859308, −12.03146560748369, −11.27908076545797, −10.83264621123931, −10.21526498246901, −9.588653303869425, −9.316083047941287, −8.867346898986310, −8.124183421475634, −7.773938401131405, −7.211082248116172, −6.668653661838732, −5.629798232224785, −5.335059088777937, −4.311959596785050, −4.022660920374979, −3.352473061514130, −2.278446024258658, −1.876287810432291, −0.8971934567165623, 0,
0.8971934567165623, 1.876287810432291, 2.278446024258658, 3.352473061514130, 4.022660920374979, 4.311959596785050, 5.335059088777937, 5.629798232224785, 6.668653661838732, 7.211082248116172, 7.773938401131405, 8.124183421475634, 8.867346898986310, 9.316083047941287, 9.588653303869425, 10.21526498246901, 10.83264621123931, 11.27908076545797, 12.03146560748369, 12.62942195859308, 13.22324117788800, 13.41691052383780, 14.23812148419434, 14.58949201386420, 14.97061764078193