| L(s) = 1 | − 3·5-s + 7-s + 13-s − 3·17-s + 8·19-s + 4·25-s − 3·29-s + 4·31-s − 3·35-s − 7·37-s + 9·41-s + 8·43-s + 12·47-s + 49-s − 9·53-s − 12·59-s − 2·61-s − 3·65-s + 4·67-s − 2·73-s + 8·79-s + 6·83-s + 9·85-s − 9·89-s + 91-s − 24·95-s − 19·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.277·13-s − 0.727·17-s + 1.83·19-s + 4/5·25-s − 0.557·29-s + 0.718·31-s − 0.507·35-s − 1.15·37-s + 1.40·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 1.23·53-s − 1.56·59-s − 0.256·61-s − 0.372·65-s + 0.488·67-s − 0.234·73-s + 0.900·79-s + 0.658·83-s + 0.976·85-s − 0.953·89-s + 0.104·91-s − 2.46·95-s − 1.92·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.83843408432173, −13.48344537902915, −12.56222147263396, −12.25674352971321, −11.98976679997387, −11.24983376967566, −10.97656639953779, −10.72699978114193, −9.752997363312913, −9.390572774635976, −8.867330641358901, −8.315959981101587, −7.737126196159740, −7.510310829232515, −7.024302326987429, −6.336742170985240, −5.661029640259309, −5.223893725858151, −4.434377911302188, −4.195885979911894, −3.522902591086382, −2.997550172967255, −2.358803032908928, −1.413739223201170, −0.8332043708359168, 0,
0.8332043708359168, 1.413739223201170, 2.358803032908928, 2.997550172967255, 3.522902591086382, 4.195885979911894, 4.434377911302188, 5.223893725858151, 5.661029640259309, 6.336742170985240, 7.024302326987429, 7.510310829232515, 7.737126196159740, 8.315959981101587, 8.867330641358901, 9.390572774635976, 9.752997363312913, 10.72699978114193, 10.97656639953779, 11.24983376967566, 11.98976679997387, 12.25674352971321, 12.56222147263396, 13.48344537902915, 13.83843408432173
