| L(s) = 1 | + 2·5-s + 2·7-s − 3·9-s + 13-s − 6·17-s + 6·19-s + 8·23-s − 25-s − 2·29-s + 10·31-s + 4·35-s − 6·37-s + 6·41-s − 4·43-s − 6·45-s − 2·47-s − 3·49-s + 6·53-s − 10·59-s + 2·61-s − 6·63-s + 2·65-s + 10·67-s + 10·71-s − 2·73-s + 4·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 9-s + 0.277·13-s − 1.45·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s − 1.30·59-s + 0.256·61-s − 0.755·63-s + 0.248·65-s + 1.22·67-s + 1.18·71-s − 0.234·73-s + 0.450·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534789142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534789142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.188026513646907960045374784175, −7.29514588164529131769968986411, −6.54793186074558134850205984114, −5.90310277112433722968505568704, −5.11930925413247931038170489747, −4.72103064086739072209808883966, −3.47250340005667196587112302866, −2.67909090540963362843709927226, −1.89806035968287884320540749063, −0.845147526495205652370721052284,
0.845147526495205652370721052284, 1.89806035968287884320540749063, 2.67909090540963362843709927226, 3.47250340005667196587112302866, 4.72103064086739072209808883966, 5.11930925413247931038170489747, 5.90310277112433722968505568704, 6.54793186074558134850205984114, 7.29514588164529131769968986411, 8.188026513646907960045374784175
