| L(s) = 1 | − 2-s − 3-s + 4-s + 2.08·5-s + 6-s + 1.85·7-s − 8-s + 9-s − 2.08·10-s − 3.25·11-s − 12-s − 13-s − 1.85·14-s − 2.08·15-s + 16-s − 0.960·17-s − 18-s − 4.14·19-s + 2.08·20-s − 1.85·21-s + 3.25·22-s − 5.98·23-s + 24-s − 0.669·25-s + 26-s − 27-s + 1.85·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.930·5-s + 0.408·6-s + 0.700·7-s − 0.353·8-s + 0.333·9-s − 0.658·10-s − 0.981·11-s − 0.288·12-s − 0.277·13-s − 0.495·14-s − 0.537·15-s + 0.250·16-s − 0.232·17-s − 0.235·18-s − 0.950·19-s + 0.465·20-s − 0.404·21-s + 0.694·22-s − 1.24·23-s + 0.204·24-s − 0.133·25-s + 0.196·26-s − 0.192·27-s + 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 17 | \( 1 + 0.960T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 + 5.98T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.406T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 0.674T + 71T^{2} \) |
| 73 | \( 1 + 1.62T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−7.69573853020843935480847876587, −6.69255418456969329638303857985, −6.15058465249414967739644742881, −5.55650785764931642561352500443, −4.79742140941333349648626561900, −4.08176417156300387304986895548, −2.60961374137913152792609410503, −2.18412568188417771410019157623, −1.20925607585711618526468635167, 0,
1.20925607585711618526468635167, 2.18412568188417771410019157623, 2.60961374137913152792609410503, 4.08176417156300387304986895548, 4.79742140941333349648626561900, 5.55650785764931642561352500443, 6.15058465249414967739644742881, 6.69255418456969329638303857985, 7.69573853020843935480847876587
