| L(s) = 1 | + 1.52·3-s − 5-s − 0.329·7-s − 0.670·9-s − 0.879·11-s + 1.35·13-s − 1.52·15-s + 5.60·17-s + 19-s − 0.502·21-s − 8.77·23-s + 25-s − 5.60·27-s + 2.54·29-s + 0.394·31-s − 1.34·33-s + 0.329·35-s − 9.34·37-s + 2.06·39-s − 3.05·41-s + 4.22·43-s + 0.670·45-s − 1.51·47-s − 6.89·49-s + 8.54·51-s − 0.669·53-s + 0.879·55-s + ⋯ |
| L(s) = 1 | + 0.881·3-s − 0.447·5-s − 0.124·7-s − 0.223·9-s − 0.265·11-s + 0.375·13-s − 0.394·15-s + 1.35·17-s + 0.229·19-s − 0.109·21-s − 1.82·23-s + 0.200·25-s − 1.07·27-s + 0.473·29-s + 0.0707·31-s − 0.233·33-s + 0.0556·35-s − 1.53·37-s + 0.330·39-s − 0.476·41-s + 0.643·43-s + 0.100·45-s − 0.220·47-s − 0.984·49-s + 1.19·51-s − 0.0919·53-s + 0.118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 7 | \( 1 + 0.329T + 7T^{2} \) |
| 11 | \( 1 + 0.879T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 23 | \( 1 + 8.77T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 - 0.394T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 0.669T + 53T^{2} \) |
| 59 | \( 1 - 0.155T + 59T^{2} \) |
| 61 | \( 1 + 9.95T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 - 0.445T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.220T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−7.87303449805076880241735310206, −7.26898704122091624497816787165, −6.24288504002892990344309046434, −5.63735597582563053788762060731, −4.75812395220511757543742658843, −3.69664300038527487013035326857, −3.36738517269404067053683234165, −2.44897756288089048277588429919, −1.45578696110106569761897222559, 0,
1.45578696110106569761897222559, 2.44897756288089048277588429919, 3.36738517269404067053683234165, 3.69664300038527487013035326857, 4.75812395220511757543742658843, 5.63735597582563053788762060731, 6.24288504002892990344309046434, 7.26898704122091624497816787165, 7.87303449805076880241735310206
