L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3.48·7-s − 8-s + 9-s − 10-s + 12-s − 5.02·13-s − 3.48·14-s + 15-s + 16-s + 3.77·17-s − 18-s − 5.39·19-s + 20-s + 3.48·21-s − 6.49·23-s − 24-s + 25-s + 5.02·26-s + 27-s + 3.48·28-s − 2.86·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.31·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.39·13-s − 0.931·14-s + 0.258·15-s + 0.250·16-s + 0.915·17-s − 0.235·18-s − 1.23·19-s + 0.223·20-s + 0.760·21-s − 1.35·23-s − 0.204·24-s + 0.200·25-s + 0.985·26-s + 0.192·27-s + 0.658·28-s − 0.532·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.086941906\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.086941906\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 3.48T + 7T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 5.39T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 0.0725T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 3.32T + 89T^{2} \) |
| 97 | \( 1 - 6.73T + 97T^{2} \) |
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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
−8.323139971463985613674353844308, −8.011133677922420463217933852765, −7.38870934793527509232498786380, −6.44841153339581171299909610465, −5.57780076198255618771927649782, −4.71188221064685597850979841091, −3.93351556969401226757587831173, −2.42563515821397385390859168681, −2.19404113899020358772371896376, −0.934104173103604681196392331456,
0.934104173103604681196392331456, 2.19404113899020358772371896376, 2.42563515821397385390859168681, 3.93351556969401226757587831173, 4.71188221064685597850979841091, 5.57780076198255618771927649782, 6.44841153339581171299909610465, 7.38870934793527509232498786380, 8.011133677922420463217933852765, 8.323139971463985613674353844308