| L(s) = 1 | + 3·2-s + 4-s − 3·5-s − 21·8-s − 9·10-s + 15·11-s + 64·13-s − 71·16-s + 84·17-s + 16·19-s − 3·20-s + 45·22-s + 84·23-s − 116·25-s + 192·26-s + 297·29-s + 253·31-s − 45·32-s + 252·34-s − 316·37-s + 48·38-s + 63·40-s + 360·41-s + 26·43-s + 15·44-s + 252·46-s − 30·47-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.268·5-s − 0.928·8-s − 0.284·10-s + 0.411·11-s + 1.36·13-s − 1.10·16-s + 1.19·17-s + 0.193·19-s − 0.0335·20-s + 0.436·22-s + 0.761·23-s − 0.927·25-s + 1.44·26-s + 1.90·29-s + 1.46·31-s − 0.248·32-s + 1.27·34-s − 1.40·37-s + 0.204·38-s + 0.249·40-s + 1.37·41-s + 0.0922·43-s + 0.0513·44-s + 0.807·46-s − 0.0931·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.072578329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072578329\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 64 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 297 T + p^{3} T^{2} \) |
| 31 | \( 1 - 253 T + p^{3} T^{2} \) |
| 37 | \( 1 + 316 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 26 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 + 363 T + p^{3} T^{2} \) |
| 59 | \( 1 + 15 T + p^{3} T^{2} \) |
| 61 | \( 1 - 118 T + p^{3} T^{2} \) |
| 67 | \( 1 + 370 T + p^{3} T^{2} \) |
| 71 | \( 1 - 342 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 467 T + p^{3} T^{2} \) |
| 83 | \( 1 - 477 T + p^{3} T^{2} \) |
| 89 | \( 1 - 906 T + p^{3} T^{2} \) |
| 97 | \( 1 + 503 T + p^{3} T^{2} \) |
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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
−10.92741559891106202143646147673, −9.840245874034159708521203948657, −8.817469973780390783242943074661, −8.003616517465058048823210060680, −6.61482869831534676779707879266, −5.87080526465409933256741657776, −4.81014610877187189424899088866, −3.82275054002836987472587538824, −2.98851310927051577184490502479, −1.02176902159605660452794947887,
1.02176902159605660452794947887, 2.98851310927051577184490502479, 3.82275054002836987472587538824, 4.81014610877187189424899088866, 5.87080526465409933256741657776, 6.61482869831534676779707879266, 8.003616517465058048823210060680, 8.817469973780390783242943074661, 9.840245874034159708521203948657, 10.92741559891106202143646147673
