| L(s) = 1 | − 2.59·2-s − 1.81·3-s + 4.70·4-s + 5-s + 4.69·6-s + 3.39·7-s − 7.01·8-s + 0.284·9-s − 2.59·10-s − 1.32·11-s − 8.53·12-s − 3.86·13-s − 8.79·14-s − 1.81·15-s + 8.75·16-s + 3.76·17-s − 0.737·18-s + 6.58·19-s + 4.70·20-s − 6.15·21-s + 3.43·22-s + 7.28·23-s + 12.7·24-s + 25-s + 10.0·26-s + 4.92·27-s + 15.9·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 1.04·3-s + 2.35·4-s + 0.447·5-s + 1.91·6-s + 1.28·7-s − 2.48·8-s + 0.0949·9-s − 0.819·10-s − 0.399·11-s − 2.46·12-s − 1.07·13-s − 2.35·14-s − 0.467·15-s + 2.18·16-s + 0.913·17-s − 0.173·18-s + 1.51·19-s + 1.05·20-s − 1.34·21-s + 0.731·22-s + 1.51·23-s + 2.59·24-s + 0.200·25-s + 1.96·26-s + 0.947·27-s + 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9128821076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9128821076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 - 3.39T + 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 0.850T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−7.991636840497283080065284838787, −7.21959436700478919860832974230, −6.79644694714380759944822282697, −5.81500315709013014230571203888, −5.22483232932411489803065647791, −4.72141414783303070299657631340, −2.92748985039974177817132339438, −2.41733503205404376489449123911, −1.07458476689875546652984274168, −0.879416462193518753233134163839,
0.879416462193518753233134163839, 1.07458476689875546652984274168, 2.41733503205404376489449123911, 2.92748985039974177817132339438, 4.72141414783303070299657631340, 5.22483232932411489803065647791, 5.81500315709013014230571203888, 6.79644694714380759944822282697, 7.21959436700478919860832974230, 7.991636840497283080065284838787
