| L(s) = 1 | + 2-s + 3-s + 4-s − 1.59·5-s + 6-s + 1.20·7-s + 8-s + 9-s − 1.59·10-s + 4.80·11-s + 12-s + 1.05·13-s + 1.20·14-s − 1.59·15-s + 16-s − 2.39·17-s + 18-s + 6.34·19-s − 1.59·20-s + 1.20·21-s + 4.80·22-s + 23-s + 24-s − 2.46·25-s + 1.05·26-s + 27-s + 1.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.711·5-s + 0.408·6-s + 0.456·7-s + 0.353·8-s + 0.333·9-s − 0.503·10-s + 1.44·11-s + 0.288·12-s + 0.292·13-s + 0.322·14-s − 0.410·15-s + 0.250·16-s − 0.580·17-s + 0.235·18-s + 1.45·19-s − 0.355·20-s + 0.263·21-s + 1.02·22-s + 0.208·23-s + 0.204·24-s − 0.493·25-s + 0.206·26-s + 0.192·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.051980884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.051980884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 8.28T + 47T^{2} \) |
| 53 | \( 1 - 1.92T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 - 1.48T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 + 3.40T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 4.46T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.297665787189093524887443649523, −7.66754123615212998188160495591, −7.02807550578153103591826098208, −6.27985531818846077060056030511, −5.35262137312306090950320880047, −4.45931099500903418374435712032, −3.82786914155815925886099028636, −3.26204302429088411895816158371, −2.07704248846980944824050433434, −1.09600211832992946758107468033,
1.09600211832992946758107468033, 2.07704248846980944824050433434, 3.26204302429088411895816158371, 3.82786914155815925886099028636, 4.45931099500903418374435712032, 5.35262137312306090950320880047, 6.27985531818846077060056030511, 7.02807550578153103591826098208, 7.66754123615212998188160495591, 8.297665787189093524887443649523
