| L(s) = 1 | + 8·3-s + 16·7-s + 37·9-s + 40·11-s + 50·13-s + 30·17-s − 40·19-s + 128·21-s + 48·23-s + 80·27-s − 34·29-s − 320·31-s + 320·33-s − 310·37-s + 400·39-s + 410·41-s + 152·43-s − 416·47-s − 87·49-s + 240·51-s + 410·53-s − 320·57-s + 200·59-s + 30·61-s + 592·63-s + 776·67-s + 384·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.863·7-s + 1.37·9-s + 1.09·11-s + 1.06·13-s + 0.428·17-s − 0.482·19-s + 1.33·21-s + 0.435·23-s + 0.570·27-s − 0.217·29-s − 1.85·31-s + 1.68·33-s − 1.37·37-s + 1.64·39-s + 1.56·41-s + 0.539·43-s − 1.29·47-s − 0.253·49-s + 0.658·51-s + 1.06·53-s − 0.743·57-s + 0.441·59-s + 0.0629·61-s + 1.18·63-s + 1.41·67-s + 0.669·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.572527100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.572527100\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 + 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 416 T + p^{3} T^{2} \) |
| 53 | \( 1 - 410 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 - 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−9.500063986950754882251664983094, −8.946696934215502822814021631729, −8.312263393578806139327757745640, −7.55044132704013839292442313112, −6.59784340661538159549951933926, −5.34489347802331128251369776429, −4.01429101447137920022477466250, −3.50765195689488865882793651829, −2.13780006390465741062937893254, −1.28609364578193050044790091343,
1.28609364578193050044790091343, 2.13780006390465741062937893254, 3.50765195689488865882793651829, 4.01429101447137920022477466250, 5.34489347802331128251369776429, 6.59784340661538159549951933926, 7.55044132704013839292442313112, 8.312263393578806139327757745640, 8.946696934215502822814021631729, 9.500063986950754882251664983094
