L(s) = 1 | − 2-s + 4-s − 1.17·5-s + 0.506·7-s − 8-s + 1.17·10-s − 3.88·11-s − 3.61·13-s − 0.506·14-s + 16-s + 6.95·17-s − 2.33·19-s − 1.17·20-s + 3.88·22-s − 1.72·23-s − 3.62·25-s + 3.61·26-s + 0.506·28-s + 4.21·29-s − 3.29·31-s − 32-s − 6.95·34-s − 0.592·35-s − 7.04·37-s + 2.33·38-s + 1.17·40-s + 10.5·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.523·5-s + 0.191·7-s − 0.353·8-s + 0.370·10-s − 1.17·11-s − 1.00·13-s − 0.135·14-s + 0.250·16-s + 1.68·17-s − 0.536·19-s − 0.261·20-s + 0.827·22-s − 0.359·23-s − 0.725·25-s + 0.708·26-s + 0.0957·28-s + 0.783·29-s − 0.591·31-s − 0.176·32-s − 1.19·34-s − 0.100·35-s − 1.15·37-s + 0.379·38-s + 0.185·40-s + 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7293389437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7293389437\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 0.506T + 7T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 + 2.33T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 6.84T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 + 0.923T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 + 8.00T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−7.87404503084185260628619242353, −7.42383739945638428496605581322, −6.66879578724653918465213848597, −5.64564112992254072604169051498, −5.21472088478674009718901702141, −4.26377024393714353493166753318, −3.33974791175690850052360831926, −2.59851873277657390121905435149, −1.71345715946999462378600293517, −0.45838413417870085481505307293,
0.45838413417870085481505307293, 1.71345715946999462378600293517, 2.59851873277657390121905435149, 3.33974791175690850052360831926, 4.26377024393714353493166753318, 5.21472088478674009718901702141, 5.64564112992254072604169051498, 6.66879578724653918465213848597, 7.42383739945638428496605581322, 7.87404503084185260628619242353