L(s) = 1 | − 1.73·3-s − 5-s + 4.18·11-s + 1.58·13-s + 1.73·15-s + 7.24·17-s + 4.89·19-s − 1.01·23-s + 25-s + 5.19·27-s + 29-s + 5.91·31-s − 7.24·33-s + 4.24·37-s − 2.74·39-s − 1.41·41-s − 6.92·43-s + 10.0·47-s − 12.5·51-s + 8.24·53-s − 4.18·55-s − 8.48·57-s − 12.2·59-s + 11.6·61-s − 1.58·65-s − 3.88·67-s + 1.75·69-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 0.447·5-s + 1.26·11-s + 0.439·13-s + 0.447·15-s + 1.75·17-s + 1.12·19-s − 0.211·23-s + 0.200·25-s + 1.00·27-s + 0.185·29-s + 1.06·31-s − 1.26·33-s + 0.697·37-s − 0.439·39-s − 0.220·41-s − 1.05·43-s + 1.47·47-s − 1.75·51-s + 1.13·53-s − 0.563·55-s − 1.12·57-s − 1.59·59-s + 1.49·61-s − 0.196·65-s − 0.474·67-s + 0.211·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.607584099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607584099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 7.24T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 7.64T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−7.71927804893679497244663482146, −7.13794600556368138183000346010, −6.29495342467085685049916288411, −5.84373010090480510412345159527, −5.16739989736761789583956187747, −4.35124001968516584950830861879, −3.56682381400269308747653576063, −2.88458943506907967997975740273, −1.33822132293943203929900533676, −0.77392431152625990269254932560,
0.77392431152625990269254932560, 1.33822132293943203929900533676, 2.88458943506907967997975740273, 3.56682381400269308747653576063, 4.35124001968516584950830861879, 5.16739989736761789583956187747, 5.84373010090480510412345159527, 6.29495342467085685049916288411, 7.13794600556368138183000346010, 7.71927804893679497244663482146