| L(s) = 1 | − 2-s − 7·4-s − 5·5-s + 15·8-s + 5·10-s − 12·11-s + 78·13-s + 41·16-s − 94·17-s − 40·19-s + 35·20-s + 12·22-s − 32·23-s + 25·25-s − 78·26-s + 50·29-s + 248·31-s − 161·32-s + 94·34-s − 434·37-s + 40·38-s − 75·40-s + 402·41-s − 68·43-s + 84·44-s + 32·46-s + 536·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.447·5-s + 0.662·8-s + 0.158·10-s − 0.328·11-s + 1.66·13-s + 0.640·16-s − 1.34·17-s − 0.482·19-s + 0.391·20-s + 0.116·22-s − 0.290·23-s + 1/5·25-s − 0.588·26-s + 0.320·29-s + 1.43·31-s − 0.889·32-s + 0.474·34-s − 1.92·37-s + 0.170·38-s − 0.296·40-s + 1.53·41-s − 0.241·43-s + 0.287·44-s + 0.102·46-s + 1.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9737832716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9737832716\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 32 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 434 T + p^{3} T^{2} \) |
| 41 | \( 1 - 402 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 536 T + p^{3} T^{2} \) |
| 53 | \( 1 + 22 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 278 T + p^{3} T^{2} \) |
| 67 | \( 1 + 164 T + p^{3} T^{2} \) |
| 71 | \( 1 + 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 82 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 + 448 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1026 T + p^{3} T^{2} \) |
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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
−8.630135586866504646874571269571, −8.255380596390110776617625321040, −7.27594444739773378484382018039, −6.36745860647977370645272570860, −5.55246319181585683990065067670, −4.40670105596695358368564170350, −4.06772799115557726027685884572, −2.91461384840793000932734081712, −1.55090301299083091585486177508, −0.49638613684054661020272717400,
0.49638613684054661020272717400, 1.55090301299083091585486177508, 2.91461384840793000932734081712, 4.06772799115557726027685884572, 4.40670105596695358368564170350, 5.55246319181585683990065067670, 6.36745860647977370645272570860, 7.27594444739773378484382018039, 8.255380596390110776617625321040, 8.630135586866504646874571269571
