L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3.49·7-s − 8-s + 9-s + 10-s − 2.98·11-s + 12-s + 0.0262·13-s − 3.49·14-s − 15-s + 16-s − 18-s − 2.12·19-s − 20-s + 3.49·21-s + 2.98·22-s − 7.10·23-s − 24-s + 25-s − 0.0262·26-s + 27-s + 3.49·28-s − 6.04·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.32·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.899·11-s + 0.288·12-s + 0.00728·13-s − 0.935·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s − 0.488·19-s − 0.223·20-s + 0.763·21-s + 0.636·22-s − 1.48·23-s − 0.204·24-s + 0.200·25-s − 0.00515·26-s + 0.192·27-s + 0.661·28-s − 1.12·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.682793249\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.682793249\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.0262T + 13T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 7.10T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 - 5.81T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 0.962T + 59T^{2} \) |
| 61 | \( 1 - 4.71T + 61T^{2} \) |
| 67 | \( 1 - 9.85T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.902412635874880132771050771930, −7.56777965945568133826717472202, −6.52983725492909973450289609484, −5.78312995655614778985021959975, −4.85651753145285467253170168770, −4.27910070891557487792285122837, −3.37483770701881890611050228026, −2.34596946137657135581941115256, −1.87300909472022850040694005863, −0.67459278473721679605110151210,
0.67459278473721679605110151210, 1.87300909472022850040694005863, 2.34596946137657135581941115256, 3.37483770701881890611050228026, 4.27910070891557487792285122837, 4.85651753145285467253170168770, 5.78312995655614778985021959975, 6.52983725492909973450289609484, 7.56777965945568133826717472202, 7.902412635874880132771050771930