L(s) = 1 | − 5-s + 7-s − 2·11-s − 6·13-s − 4·17-s + 2·19-s − 23-s + 25-s − 8·29-s − 10·31-s − 35-s − 2·37-s + 10·41-s + 10·43-s − 6·47-s + 49-s + 10·53-s + 2·55-s − 12·59-s + 10·61-s + 6·65-s − 14·67-s − 6·71-s − 10·73-s − 2·77-s + 8·79-s − 14·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.66·13-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.48·29-s − 1.79·31-s − 0.169·35-s − 0.328·37-s + 1.56·41-s + 1.52·43-s − 0.875·47-s + 1/7·49-s + 1.37·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + 0.744·65-s − 1.71·67-s − 0.712·71-s − 1.17·73-s − 0.227·77-s + 0.900·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.21963048373073, −13.52713022810063, −13.03813466594022, −12.63754838840542, −12.20417388649865, −11.62565260652595, −11.17000611666664, −10.76536538590037, −10.27992121442086, −9.582761975496301, −9.172472036432016, −8.834003943382529, −7.955574264864887, −7.580878876047267, −7.289161042337290, −6.824551999245544, −5.820562740860333, −5.535653932927414, −4.980193834841711, −4.274389343355406, −4.041825438206085, −3.115589512044038, −2.524977306289677, −2.071064287791647, −1.277913009754535, 0, 0,
1.277913009754535, 2.071064287791647, 2.524977306289677, 3.115589512044038, 4.041825438206085, 4.274389343355406, 4.980193834841711, 5.535653932927414, 5.820562740860333, 6.824551999245544, 7.289161042337290, 7.580878876047267, 7.955574264864887, 8.834003943382529, 9.172472036432016, 9.582761975496301, 10.27992121442086, 10.76536538590037, 11.17000611666664, 11.62565260652595, 12.20417388649865, 12.63754838840542, 13.03813466594022, 13.52713022810063, 14.21963048373073