| L(s) = 1 | − 0.417·2-s − 5.36·3-s − 7.82·4-s − 14.2·5-s + 2.23·6-s − 7.39·7-s + 6.59·8-s + 1.78·9-s + 5.92·10-s + 41.9·12-s + 49.4·13-s + 3.08·14-s + 76.2·15-s + 59.8·16-s + 17·17-s − 0.744·18-s − 91.9·19-s + 111.·20-s + 39.6·21-s + 142.·23-s − 35.4·24-s + 76.7·25-s − 20.6·26-s + 135.·27-s + 57.9·28-s − 165.·29-s − 31.7·30-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 1.03·3-s − 0.978·4-s − 1.27·5-s + 0.152·6-s − 0.399·7-s + 0.291·8-s + 0.0660·9-s + 0.187·10-s + 1.01·12-s + 1.05·13-s + 0.0588·14-s + 1.31·15-s + 0.935·16-s + 0.242·17-s − 0.00974·18-s − 1.11·19-s + 1.24·20-s + 0.412·21-s + 1.29·23-s − 0.301·24-s + 0.614·25-s − 0.155·26-s + 0.964·27-s + 0.390·28-s − 1.06·29-s − 0.193·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05878580867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05878580867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 0.417T + 8T^{2} \) |
| 3 | \( 1 + 5.36T + 27T^{2} \) |
| 5 | \( 1 + 14.2T + 125T^{2} \) |
| 7 | \( 1 + 7.39T + 343T^{2} \) |
| 13 | \( 1 - 49.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 91.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 333.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 688.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 535.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 496.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 610.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 863.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 546.T + 9.12e5T^{2} \) |
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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
−8.753574187296617249809434551530, −8.103572817861144546279015442906, −7.22004392012102714201751752429, −6.35772273107802195900854850924, −5.49582837334894184081267410349, −4.78392134756250679295234371543, −3.86178565422109597920173845101, −3.31230171118040444585298825771, −1.35000556133758427926744779911, −0.13214311882466097338603684040,
0.13214311882466097338603684040, 1.35000556133758427926744779911, 3.31230171118040444585298825771, 3.86178565422109597920173845101, 4.78392134756250679295234371543, 5.49582837334894184081267410349, 6.35772273107802195900854850924, 7.22004392012102714201751752429, 8.103572817861144546279015442906, 8.753574187296617249809434551530
