| L(s) = 1 | + 2·3-s − 7-s + 9-s + 5·11-s + 13-s − 4·17-s − 7·19-s − 2·21-s + 23-s − 4·27-s + 5·29-s − 2·31-s + 10·33-s − 2·37-s + 2·39-s + 11·41-s − 43-s + 8·47-s − 6·49-s − 8·51-s − 14·57-s + 14·59-s + 10·61-s − 63-s + 8·67-s + 2·69-s + 10·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.970·17-s − 1.60·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.928·29-s − 0.359·31-s + 1.74·33-s − 0.328·37-s + 0.320·39-s + 1.71·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s − 1.12·51-s − 1.85·57-s + 1.82·59-s + 1.28·61-s − 0.125·63-s + 0.977·67-s + 0.240·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.116023106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.116023106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.891692787276342096834772565072, −6.89683672326933800534947165278, −6.56191559535968493178322265131, −5.85320262180626887736682335254, −4.70877365547697582315761730393, −3.92338595003306067616288650284, −3.62349190950048953286603490312, −2.48078906059132474481242328022, −2.05191779631866543865602232431, −0.793767924709225545348060160981,
0.793767924709225545348060160981, 2.05191779631866543865602232431, 2.48078906059132474481242328022, 3.62349190950048953286603490312, 3.92338595003306067616288650284, 4.70877365547697582315761730393, 5.85320262180626887736682335254, 6.56191559535968493178322265131, 6.89683672326933800534947165278, 7.891692787276342096834772565072
