| L(s) = 1 | − 2-s − 0.698·3-s + 4-s − 2.43·5-s + 0.698·6-s + 7-s − 8-s − 2.51·9-s + 2.43·10-s − 0.698·12-s − 2.86·13-s − 14-s + 1.69·15-s + 16-s − 1.90·17-s + 2.51·18-s − 19-s − 2.43·20-s − 0.698·21-s + 6.50·23-s + 0.698·24-s + 0.909·25-s + 2.86·26-s + 3.85·27-s + 28-s + 7.02·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.403·3-s + 0.5·4-s − 1.08·5-s + 0.285·6-s + 0.377·7-s − 0.353·8-s − 0.837·9-s + 0.768·10-s − 0.201·12-s − 0.793·13-s − 0.267·14-s + 0.438·15-s + 0.250·16-s − 0.463·17-s + 0.591·18-s − 0.229·19-s − 0.543·20-s − 0.152·21-s + 1.35·23-s + 0.142·24-s + 0.181·25-s + 0.561·26-s + 0.741·27-s + 0.188·28-s + 1.30·29-s − 0.310·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.698T + 3T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 0.383T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 + 5.17T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 + 0.348T + 53T^{2} \) |
| 59 | \( 1 + 1.62T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.234999307339322360131246502665, −7.25070775636492239351834471115, −6.75718347340221208926542011171, −5.84991464813146309409342755014, −4.92510652540056605999537513745, −4.33528591138150974948436921344, −3.13009240365442165388728266235, −2.47328869430187173104693663807, −1.00975746967924539269192426675, 0,
1.00975746967924539269192426675, 2.47328869430187173104693663807, 3.13009240365442165388728266235, 4.33528591138150974948436921344, 4.92510652540056605999537513745, 5.84991464813146309409342755014, 6.75718347340221208926542011171, 7.25070775636492239351834471115, 8.234999307339322360131246502665
