| L(s) = 1 | − 2-s + 4-s − 5-s + 4.94·7-s − 8-s − 3·9-s + 10-s + 1.53·11-s − 6.10·13-s − 4.94·14-s + 16-s + 3·18-s + 2.06·19-s − 20-s − 1.53·22-s + 3.41·23-s + 25-s + 6.10·26-s + 4.94·28-s + 5.06·29-s − 7.75·31-s − 32-s − 4.94·35-s − 3·36-s + 5.45·37-s − 2.06·38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.86·7-s − 0.353·8-s − 9-s + 0.316·10-s + 0.461·11-s − 1.69·13-s − 1.32·14-s + 0.250·16-s + 0.707·18-s + 0.473·19-s − 0.223·20-s − 0.326·22-s + 0.711·23-s + 0.200·25-s + 1.19·26-s + 0.934·28-s + 0.940·29-s − 1.39·31-s − 0.176·32-s − 0.835·35-s − 0.5·36-s + 0.896·37-s − 0.334·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284319680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284319680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 19 | \( 1 - 2.06T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 - 5.45T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 7.43T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 5.63T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 4.77T + 73T^{2} \) |
| 79 | \( 1 + 0.822T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−8.611670939616779495454382991846, −8.121686670869563413450353834520, −7.42528304240884886277313785718, −6.85580583043543374421733785123, −5.46587052527886151475516131737, −5.07525262268060322140295282819, −4.09371123750513514137666890756, −2.80936818662686950643589007560, −2.00420659734479728744135953771, −0.77859497909420671877121523381,
0.77859497909420671877121523381, 2.00420659734479728744135953771, 2.80936818662686950643589007560, 4.09371123750513514137666890756, 5.07525262268060322140295282819, 5.46587052527886151475516131737, 6.85580583043543374421733785123, 7.42528304240884886277313785718, 8.121686670869563413450353834520, 8.611670939616779495454382991846
