| L(s) = 1 | + 26·7-s − 45·11-s + 44·13-s − 117·17-s − 91·19-s + 18·23-s − 144·29-s + 26·31-s − 214·37-s + 459·41-s − 460·43-s + 468·47-s + 333·49-s − 558·53-s + 72·59-s − 118·61-s + 251·67-s − 108·71-s + 299·73-s − 1.17e3·77-s − 898·79-s − 927·83-s − 351·89-s + 1.14e3·91-s + 386·97-s + 954·101-s − 772·103-s + ⋯ |
| L(s) = 1 | + 1.40·7-s − 1.23·11-s + 0.938·13-s − 1.66·17-s − 1.09·19-s + 0.163·23-s − 0.922·29-s + 0.150·31-s − 0.950·37-s + 1.74·41-s − 1.63·43-s + 1.45·47-s + 0.970·49-s − 1.44·53-s + 0.158·59-s − 0.247·61-s + 0.457·67-s − 0.180·71-s + 0.479·73-s − 1.73·77-s − 1.27·79-s − 1.22·83-s − 0.418·89-s + 1.31·91-s + 0.404·97-s + 0.939·101-s − 0.738·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 117 T + p^{3} T^{2} \) |
| 19 | \( 1 + 91 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 26 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 459 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 - 468 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 72 T + p^{3} T^{2} \) |
| 61 | \( 1 + 118 T + p^{3} T^{2} \) |
| 67 | \( 1 - 251 T + p^{3} T^{2} \) |
| 71 | \( 1 + 108 T + p^{3} T^{2} \) |
| 73 | \( 1 - 299 T + p^{3} T^{2} \) |
| 79 | \( 1 + 898 T + p^{3} T^{2} \) |
| 83 | \( 1 + 927 T + p^{3} T^{2} \) |
| 89 | \( 1 + 351 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−9.067097859850360137490642193981, −8.435301088174339859196192379885, −7.78564118842372946995127301916, −6.75990995819829741589509138519, −5.71262902248282020416523314000, −4.81616111022271634887205969740, −4.04755402334840306342624864315, −2.52257418223036102878174608543, −1.62131451557060714015625674328, 0,
1.62131451557060714015625674328, 2.52257418223036102878174608543, 4.04755402334840306342624864315, 4.81616111022271634887205969740, 5.71262902248282020416523314000, 6.75990995819829741589509138519, 7.78564118842372946995127301916, 8.435301088174339859196192379885, 9.067097859850360137490642193981
