| L(s) = 1 | + 3-s − 7-s + 9-s + 5·13-s − 2·17-s − 19-s − 21-s + 8·23-s + 27-s − 10·29-s − 7·31-s + 37-s + 5·39-s + 6·41-s + 11·43-s − 6·47-s + 49-s − 2·51-s + 2·53-s − 57-s − 6·59-s + 13·61-s − 63-s − 3·67-s + 8·69-s − 12·71-s − 11·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.38·13-s − 0.485·17-s − 0.229·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.85·29-s − 1.25·31-s + 0.164·37-s + 0.800·39-s + 0.937·41-s + 1.67·43-s − 0.875·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.132·57-s − 0.781·59-s + 1.66·61-s − 0.125·63-s − 0.366·67-s + 0.963·69-s − 1.42·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.613662341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.613662341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.84974924485522, −12.74294040524990, −11.87599829769457, −11.21334358742281, −11.05427627059491, −10.68901423925651, −9.992256840519740, −9.415148698248851, −9.009085739901238, −8.849160572872697, −8.248613193180639, −7.576517017991457, −7.232821986865894, −6.803406770237683, −6.142685351552237, −5.663140760894540, −5.320028356387161, −4.358019936369126, −4.079733848395963, −3.573025152133022, −2.930141345465508, −2.567355560183670, −1.651160143714194, −1.336690093270416, −0.4200026050553193,
0.4200026050553193, 1.336690093270416, 1.651160143714194, 2.567355560183670, 2.930141345465508, 3.573025152133022, 4.079733848395963, 4.358019936369126, 5.320028356387161, 5.663140760894540, 6.142685351552237, 6.803406770237683, 7.232821986865894, 7.576517017991457, 8.248613193180639, 8.849160572872697, 9.009085739901238, 9.415148698248851, 9.992256840519740, 10.68901423925651, 11.05427627059491, 11.21334358742281, 11.87599829769457, 12.74294040524990, 12.84974924485522
