L(s) = 1 | − 2-s + 1.03·3-s + 4-s + 5-s − 1.03·6-s − 3.65·7-s − 8-s − 1.92·9-s − 10-s + 3.37·11-s + 1.03·12-s + 3.74·13-s + 3.65·14-s + 1.03·15-s + 16-s − 0.215·17-s + 1.92·18-s − 3.16·19-s + 20-s − 3.79·21-s − 3.37·22-s + 5.41·23-s − 1.03·24-s + 25-s − 3.74·26-s − 5.10·27-s − 3.65·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.599·3-s + 0.5·4-s + 0.447·5-s − 0.423·6-s − 1.38·7-s − 0.353·8-s − 0.640·9-s − 0.316·10-s + 1.01·11-s + 0.299·12-s + 1.03·13-s + 0.977·14-s + 0.267·15-s + 0.250·16-s − 0.0522·17-s + 0.453·18-s − 0.726·19-s + 0.223·20-s − 0.828·21-s − 0.719·22-s + 1.12·23-s − 0.211·24-s + 0.200·25-s − 0.734·26-s − 0.983·27-s − 0.691·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.567572628\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.567572628\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 1.03T + 3T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 + 0.215T + 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 0.812T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 0.202T + 67T^{2} \) |
| 71 | \( 1 - 4.07T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.430555708626147067148937439076, −7.37520610826037891041631555734, −6.53976683360787529100535536358, −6.29946697714370124988897051243, −5.49497856386639872804312740306, −4.17051402402453263587668829241, −3.35416700568729816204749858488, −2.83953277474922999598739901472, −1.82315256645434075823468289084, −0.70671604275720131854461111304,
0.70671604275720131854461111304, 1.82315256645434075823468289084, 2.83953277474922999598739901472, 3.35416700568729816204749858488, 4.17051402402453263587668829241, 5.49497856386639872804312740306, 6.29946697714370124988897051243, 6.53976683360787529100535536358, 7.37520610826037891041631555734, 8.430555708626147067148937439076