| L(s) = 1 | + 7-s − 2·13-s + 8·17-s − 19-s + 4·23-s − 5·25-s − 2·37-s − 2·41-s − 4·43-s + 6·47-s + 49-s + 12·53-s + 4·59-s − 2·61-s + 4·67-s − 6·71-s + 6·73-s + 4·79-s + 6·83-s − 18·89-s − 2·91-s + 14·97-s − 8·101-s − 8·103-s + 14·107-s + 2·109-s + 12·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.554·13-s + 1.94·17-s − 0.229·19-s + 0.834·23-s − 25-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 1.64·53-s + 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.712·71-s + 0.702·73-s + 0.450·79-s + 0.658·83-s − 1.90·89-s − 0.209·91-s + 1.42·97-s − 0.796·101-s − 0.788·103-s + 1.35·107-s + 0.191·109-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212362725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212362725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.58390096127455694469142543695, −7.21364939555399481524672923662, −6.29419328755208021740414841934, −5.46870592038837776596176063469, −5.13657558296864309882179274563, −4.13929619780105239914084667870, −3.45813910768920889637644640566, −2.62160278433278429316063358442, −1.69178674004581491616959959826, −0.73018489888737339389569784960,
0.73018489888737339389569784960, 1.69178674004581491616959959826, 2.62160278433278429316063358442, 3.45813910768920889637644640566, 4.13929619780105239914084667870, 5.13657558296864309882179274563, 5.46870592038837776596176063469, 6.29419328755208021740414841934, 7.21364939555399481524672923662, 7.58390096127455694469142543695
