| L(s) = 1 | + 2·7-s + 6·13-s − 7·17-s + 7·19-s − 7·23-s + 6·29-s + 3·31-s + 6·37-s + 4·41-s − 8·43-s + 4·47-s − 3·49-s + 5·53-s + 6·59-s − 3·61-s + 10·67-s + 12·71-s − 16·73-s + 79-s − 9·83-s − 4·89-s + 12·91-s + 16·97-s + 4·101-s − 4·103-s − 4·107-s + 5·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.66·13-s − 1.69·17-s + 1.60·19-s − 1.45·23-s + 1.11·29-s + 0.538·31-s + 0.986·37-s + 0.624·41-s − 1.21·43-s + 0.583·47-s − 3/7·49-s + 0.686·53-s + 0.781·59-s − 0.384·61-s + 1.22·67-s + 1.42·71-s − 1.87·73-s + 0.112·79-s − 0.987·83-s − 0.423·89-s + 1.25·91-s + 1.62·97-s + 0.398·101-s − 0.394·103-s − 0.386·107-s + 0.478·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384782599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384782599\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.304928948528404372459879971777, −7.56814408983346912572966783560, −6.62749150839164561852119720118, −6.10638884221662800514725828488, −5.27255777418581680532613196197, −4.42800283910545065997972215278, −3.83182362425152738477120240118, −2.79278434699131491112945311494, −1.80974043194117476126650089353, −0.873398469787543979524321981059,
0.873398469787543979524321981059, 1.80974043194117476126650089353, 2.79278434699131491112945311494, 3.83182362425152738477120240118, 4.42800283910545065997972215278, 5.27255777418581680532613196197, 6.10638884221662800514725828488, 6.62749150839164561852119720118, 7.56814408983346912572966783560, 8.304928948528404372459879971777
