| L(s) = 1 | + 3-s − 5-s − 4.60·7-s + 9-s − 2.60·11-s − 2.60·13-s − 15-s + 2·17-s + 19-s − 4.60·21-s + 2·23-s + 25-s + 27-s − 4.60·29-s − 4·31-s − 2.60·33-s + 4.60·35-s + 10.6·37-s − 2.60·39-s − 0.605·41-s − 3.39·43-s − 45-s − 6·47-s + 14.2·49-s + 2·51-s + 2.60·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.74·7-s + 0.333·9-s − 0.785·11-s − 0.722·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s − 1.00·21-s + 0.417·23-s + 0.200·25-s + 0.192·27-s − 0.855·29-s − 0.718·31-s − 0.453·33-s + 0.778·35-s + 1.74·37-s − 0.417·39-s − 0.0945·41-s − 0.517·43-s − 0.149·45-s − 0.875·47-s + 2.03·49-s + 0.280·51-s + 0.351·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184922889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184922889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 0.605T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 0.605T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−8.321577928540427717451531110982, −7.44848934001329146869859299692, −7.13601941883370660859712228513, −6.18717679152479307982043399432, −5.44044268795083101009170148971, −4.47964798320669772642356770475, −3.50023852766358592274062027944, −3.06555570997676622099149393577, −2.20091255099739029833495051919, −0.55688154813885186195067016660,
0.55688154813885186195067016660, 2.20091255099739029833495051919, 3.06555570997676622099149393577, 3.50023852766358592274062027944, 4.47964798320669772642356770475, 5.44044268795083101009170148971, 6.18717679152479307982043399432, 7.13601941883370660859712228513, 7.44848934001329146869859299692, 8.321577928540427717451531110982
