| L(s) = 1 | + 3.41·3-s + 2.33·5-s + 4.94·7-s + 8.65·9-s − 0.527·11-s − 5.30·13-s + 7.95·15-s + 4.95·17-s − 6.78·19-s + 16.8·21-s + 1.59·23-s + 0.431·25-s + 19.3·27-s − 1.58·29-s − 1.80·31-s − 1.80·33-s + 11.5·35-s − 5.53·37-s − 18.1·39-s + 4.28·41-s − 8.24·43-s + 20.1·45-s − 5.67·47-s + 17.4·49-s + 16.9·51-s − 2.22·53-s − 1.22·55-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 1.04·5-s + 1.87·7-s + 2.88·9-s − 0.158·11-s − 1.47·13-s + 2.05·15-s + 1.20·17-s − 1.55·19-s + 3.68·21-s + 0.331·23-s + 0.0863·25-s + 3.71·27-s − 0.294·29-s − 0.325·31-s − 0.313·33-s + 1.94·35-s − 0.909·37-s − 2.89·39-s + 0.669·41-s − 1.25·43-s + 3.00·45-s − 0.827·47-s + 2.49·49-s + 2.36·51-s − 0.306·53-s − 0.165·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.866944442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.866944442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 0.527T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 4.95T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−8.018216487488067783308101999837, −7.38709277881821654135160056594, −6.78485186524840651495478660664, −5.51415109442258823093807619414, −4.89066162589431421870140880235, −4.29387835826055931870192098572, −3.35616787599198041773848192829, −2.34101493836937462041053433585, −2.04909027891242531457064734863, −1.37205675444301014274267935030,
1.37205675444301014274267935030, 2.04909027891242531457064734863, 2.34101493836937462041053433585, 3.35616787599198041773848192829, 4.29387835826055931870192098572, 4.89066162589431421870140880235, 5.51415109442258823093807619414, 6.78485186524840651495478660664, 7.38709277881821654135160056594, 8.018216487488067783308101999837
