L(s) = 1 | + 5-s + 2.71·7-s − 2.71·11-s + 13-s + 2.83·17-s + 3.55·19-s − 4.83·23-s + 25-s − 6·29-s − 7.55·31-s + 2.71·35-s − 4.27·37-s − 2.83·41-s − 11.1·43-s − 11.5·47-s + 0.397·49-s − 1.16·53-s − 2.71·55-s − 2.11·59-s + 6.60·61-s + 65-s − 1.88·67-s − 6.71·71-s + 9.11·73-s − 7.39·77-s − 10.2·79-s + 2.11·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.02·7-s − 0.820·11-s + 0.277·13-s + 0.688·17-s + 0.816·19-s − 1.00·23-s + 0.200·25-s − 1.11·29-s − 1.35·31-s + 0.459·35-s − 0.703·37-s − 0.443·41-s − 1.69·43-s − 1.68·47-s + 0.0567·49-s − 0.159·53-s − 0.366·55-s − 0.275·59-s + 0.845·61-s + 0.124·65-s − 0.230·67-s − 0.797·71-s + 1.06·73-s − 0.843·77-s − 1.15·79-s + 0.232·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.45518515716597298897710960843, −6.76529691164989551310313068313, −5.75239271112804467968828778615, −5.35547693496961393329091092477, −4.79271731615888810487528844481, −3.74060216563610477590844572368, −3.09777465896630303930439162194, −1.92914235376511877405223222358, −1.51245668206700871652279007050, 0,
1.51245668206700871652279007050, 1.92914235376511877405223222358, 3.09777465896630303930439162194, 3.74060216563610477590844572368, 4.79271731615888810487528844481, 5.35547693496961393329091092477, 5.75239271112804467968828778615, 6.76529691164989551310313068313, 7.45518515716597298897710960843