| L(s) = 1 | + 2-s − 3-s + 4-s − 3.79·5-s − 6-s + 8-s + 9-s − 3.79·10-s + 11-s − 12-s − 0.361·13-s + 3.79·15-s + 16-s + 4.11·17-s + 18-s − 4.15·19-s − 3.79·20-s + 22-s + 0.542·23-s − 24-s + 9.42·25-s − 0.361·26-s − 27-s + 0.767·29-s + 3.79·30-s + 8.80·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.69·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.20·10-s + 0.301·11-s − 0.288·12-s − 0.100·13-s + 0.980·15-s + 0.250·16-s + 0.998·17-s + 0.235·18-s − 0.954·19-s − 0.849·20-s + 0.213·22-s + 0.113·23-s − 0.204·24-s + 1.88·25-s − 0.0707·26-s − 0.192·27-s + 0.142·29-s + 0.693·30-s + 1.58·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.79T + 5T^{2} \) |
| 13 | \( 1 + 0.361T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 0.542T + 23T^{2} \) |
| 29 | \( 1 - 0.767T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.603T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 0.0321T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.065783011158019704207617412615, −7.50677295350982395096519332426, −6.68007980280978078381712472922, −6.09332866244020388012702383142, −4.89916732415965219480614454452, −4.51331205113347849306128863660, −3.63536493757971842261216466783, −2.96016821421270931837700693018, −1.36829110051266523865948189964, 0,
1.36829110051266523865948189964, 2.96016821421270931837700693018, 3.63536493757971842261216466783, 4.51331205113347849306128863660, 4.89916732415965219480614454452, 6.09332866244020388012702383142, 6.68007980280978078381712472922, 7.50677295350982395096519332426, 8.065783011158019704207617412615
