L(s) = 1 | + 2.50·3-s + 4.26·5-s − 7-s + 3.26·9-s − 11-s − 1.17·13-s + 10.6·15-s − 4.58·17-s + 2.93·19-s − 2.50·21-s + 7.86·23-s + 13.2·25-s + 0.669·27-s − 3.00·29-s + 1.91·31-s − 2.50·33-s − 4.26·35-s + 9.74·37-s − 2.93·39-s + 2.77·41-s − 9.75·43-s + 13.9·45-s + 5.24·47-s + 49-s − 11.4·51-s + 3.93·53-s − 4.26·55-s + ⋯ |
L(s) = 1 | + 1.44·3-s + 1.90·5-s − 0.377·7-s + 1.08·9-s − 0.301·11-s − 0.325·13-s + 2.75·15-s − 1.11·17-s + 0.673·19-s − 0.546·21-s + 1.64·23-s + 2.64·25-s + 0.128·27-s − 0.558·29-s + 0.343·31-s − 0.435·33-s − 0.721·35-s + 1.60·37-s − 0.470·39-s + 0.432·41-s − 1.48·43-s + 2.07·45-s + 0.764·47-s + 0.142·49-s − 1.60·51-s + 0.539·53-s − 0.575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.717666665\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.717666665\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 31 | \( 1 - 1.91T + 31T^{2} \) |
| 37 | \( 1 - 9.74T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 + 9.75T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 - 0.669T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.550801468833430659503782468304, −7.51637589546885472213269845860, −6.87742126660644033571614859243, −6.15103309413195450952748572994, −5.33322249497978313606759963860, −4.60685161215476003914145371215, −3.40072442898468440756537096918, −2.60912354912003130527356194272, −2.27447681388460985207781079413, −1.19726984416850577284836703686,
1.19726984416850577284836703686, 2.27447681388460985207781079413, 2.60912354912003130527356194272, 3.40072442898468440756537096918, 4.60685161215476003914145371215, 5.33322249497978313606759963860, 6.15103309413195450952748572994, 6.87742126660644033571614859243, 7.51637589546885472213269845860, 8.550801468833430659503782468304