| L(s) = 1 | − 2-s − 3-s + 4-s + 1.23·5-s + 6-s + 0.763·7-s − 8-s + 9-s − 1.23·10-s − 11-s − 12-s − 0.763·14-s − 1.23·15-s + 16-s + 17-s − 18-s + 6.47·19-s + 1.23·20-s − 0.763·21-s + 22-s + 3.23·23-s + 24-s − 3.47·25-s − 27-s + 0.763·28-s − 3.23·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.552·5-s + 0.408·6-s + 0.288·7-s − 0.353·8-s + 0.333·9-s − 0.390·10-s − 0.301·11-s − 0.288·12-s − 0.204·14-s − 0.319·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.48·19-s + 0.276·20-s − 0.166·21-s + 0.213·22-s + 0.674·23-s + 0.204·24-s − 0.694·25-s − 0.192·27-s + 0.144·28-s − 0.600·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107281716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107281716\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 0.763T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.600665228395018300455213749135, −9.399252520007897762987754749826, −8.044097158201193939960298795862, −7.50029949658997978198515062546, −6.49054191043513535293883806551, −5.65003462906149314661885503241, −4.92336851946199228656103790313, −3.47129703918503835265965699432, −2.17761461953923587463902719531, −0.944766732297701348946355695609,
0.944766732297701348946355695609, 2.17761461953923587463902719531, 3.47129703918503835265965699432, 4.92336851946199228656103790313, 5.65003462906149314661885503241, 6.49054191043513535293883806551, 7.50029949658997978198515062546, 8.044097158201193939960298795862, 9.399252520007897762987754749826, 9.600665228395018300455213749135
