| L(s) = 1 | + 0.491·3-s − 1.41·5-s − 4.82·7-s − 2.75·9-s + 2.33·11-s + 3.75·13-s − 0.694·15-s − 0.694·19-s − 2.36·21-s + 1.99·23-s − 2.99·25-s − 2.82·27-s − 3.90·29-s + 8.63·31-s + 1.14·33-s + 6.82·35-s + 10.0·37-s + 1.84·39-s + 7.25·41-s − 10.8·43-s + 3.90·45-s + 3.51·47-s + 16.2·49-s − 8.90·53-s − 3.30·55-s − 0.341·57-s + 8.21·59-s + ⋯ |
| L(s) = 1 | + 0.283·3-s − 0.632·5-s − 1.82·7-s − 0.919·9-s + 0.704·11-s + 1.04·13-s − 0.179·15-s − 0.159·19-s − 0.517·21-s + 0.416·23-s − 0.599·25-s − 0.544·27-s − 0.724·29-s + 1.55·31-s + 0.199·33-s + 1.15·35-s + 1.65·37-s + 0.295·39-s + 1.13·41-s − 1.65·43-s + 0.581·45-s + 0.513·47-s + 2.32·49-s − 1.22·53-s − 0.445·55-s − 0.0451·57-s + 1.06·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 0.491T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 19 | \( 1 + 0.694T + 19T^{2} \) |
| 23 | \( 1 - 1.99T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 + 8.90T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 3.37T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 4.98T + 89T^{2} \) |
| 97 | \( 1 + 0.891T + 97T^{2} \) |
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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
−7.34163748155197044196601368791, −6.52080861441485132180232628421, −6.19945287329659736461774971889, −5.53144514412881936947081087144, −4.26054547318469099609621762687, −3.74559614644553429568493373684, −3.13244395656430177266677103191, −2.48286394970967258993700715228, −1.02346092651300700335309182943, 0,
1.02346092651300700335309182943, 2.48286394970967258993700715228, 3.13244395656430177266677103191, 3.74559614644553429568493373684, 4.26054547318469099609621762687, 5.53144514412881936947081087144, 6.19945287329659736461774971889, 6.52080861441485132180232628421, 7.34163748155197044196601368791
