| L(s) = 1 | − 2-s − 3-s + 4-s − 2.64·5-s + 6-s − 8-s + 9-s + 2.64·10-s + 11-s − 12-s − 4·13-s + 2.64·15-s + 16-s − 3·17-s − 18-s − 5.29·19-s − 2.64·20-s − 22-s + 2.64·23-s + 24-s + 2.00·25-s + 4·26-s − 27-s + 2·29-s − 2.64·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.18·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.836·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.683·15-s + 0.250·16-s − 0.727·17-s − 0.235·18-s − 1.21·19-s − 0.591·20-s − 0.213·22-s + 0.551·23-s + 0.204·24-s + 0.400·25-s + 0.784·26-s − 0.192·27-s + 0.371·29-s − 0.483·30-s − 0.718·31-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
| \(L(1)\) |
\(\approx\) |
\(0.3715969415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3715969415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T + 5T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 3.93T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 9.58T + 97T^{2} \) |
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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
−8.861829895023290411394302596832, −7.71964525481191952535765652756, −7.34255495612888113464009018230, −6.62516579549717356905779245327, −5.76200902751484007177084354807, −4.64998650550915022787716469638, −4.12694536639305925929978426688, −2.96577243358768784723279503753, −1.85772645634143644053923444708, −0.40411190916830400535300485387,
0.40411190916830400535300485387, 1.85772645634143644053923444708, 2.96577243358768784723279503753, 4.12694536639305925929978426688, 4.64998650550915022787716469638, 5.76200902751484007177084354807, 6.62516579549717356905779245327, 7.34255495612888113464009018230, 7.71964525481191952535765652756, 8.861829895023290411394302596832
