| L(s) = 1 | − 3-s + 9-s − 2·11-s − 13-s − 8·17-s + 7·19-s + 6·23-s − 27-s + 4·29-s + 8·31-s + 2·33-s − 7·37-s + 39-s + 2·41-s − 4·43-s − 12·47-s + 8·51-s + 4·53-s − 7·57-s + 12·59-s + 3·61-s − 9·67-s − 6·69-s − 12·71-s − 9·73-s − 17·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.94·17-s + 1.60·19-s + 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.348·33-s − 1.15·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 1.75·47-s + 1.12·51-s + 0.549·53-s − 0.927·57-s + 1.56·59-s + 0.384·61-s − 1.09·67-s − 0.722·69-s − 1.42·71-s − 1.05·73-s − 1.91·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.61973369758587, −14.95472312244727, −14.42968430043526, −13.63742848907971, −13.20825019664792, −13.01335590474500, −11.99941065122050, −11.71813346321550, −11.25938372142772, −10.57071736248951, −10.10903967429223, −9.620545597933381, −8.717671933114239, −8.563214440635029, −7.589028140856555, −7.050773688664581, −6.667767333262096, −5.938076557880303, −5.195146685073981, −4.800323051002611, −4.266831025950843, −3.173720568908117, −2.760395662999611, −1.806129302689431, −0.9316558361693093, 0,
0.9316558361693093, 1.806129302689431, 2.760395662999611, 3.173720568908117, 4.266831025950843, 4.800323051002611, 5.195146685073981, 5.938076557880303, 6.667767333262096, 7.050773688664581, 7.589028140856555, 8.563214440635029, 8.717671933114239, 9.620545597933381, 10.10903967429223, 10.57071736248951, 11.25938372142772, 11.71813346321550, 11.99941065122050, 13.01335590474500, 13.20825019664792, 13.63742848907971, 14.42968430043526, 14.95472312244727, 15.61973369758587
