| L(s) = 1 | + 2.16·3-s − 1.41·5-s + 3.14·7-s + 1.69·9-s + 0.661·11-s − 0.694·13-s − 3.06·15-s − 3.06·19-s + 6.82·21-s − 5.97·23-s − 2.99·25-s − 2.82·27-s + 2.39·29-s + 4.01·31-s + 1.43·33-s − 4.45·35-s − 9.21·37-s − 1.50·39-s − 12.0·41-s + 0.453·43-s − 2.39·45-s − 5.38·47-s + 2.91·49-s − 4.73·53-s − 0.935·55-s − 6.63·57-s + 1.67·59-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 0.632·5-s + 1.19·7-s + 0.564·9-s + 0.199·11-s − 0.192·13-s − 0.791·15-s − 0.702·19-s + 1.48·21-s − 1.24·23-s − 0.599·25-s − 0.544·27-s + 0.445·29-s + 0.720·31-s + 0.249·33-s − 0.752·35-s − 1.51·37-s − 0.240·39-s − 1.88·41-s + 0.0691·43-s − 0.357·45-s − 0.786·47-s + 0.416·49-s − 0.650·53-s − 0.126·55-s − 0.879·57-s + 0.218·59-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 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 0.661T + 11T^{2} \) |
| 13 | \( 1 + 0.694T + 13T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 0.453T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 7.48T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 - 8.95T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.75596315170035840403601785916, −6.88297244069579931175325797636, −6.08529635580399366863320012564, −5.07538788930950666598416222140, −4.46120291315624812820933360632, −3.75957777928413158754001196342, −3.12419872573209471149520953193, −2.08439158152238356770232912645, −1.62774137112587042226649121012, 0,
1.62774137112587042226649121012, 2.08439158152238356770232912645, 3.12419872573209471149520953193, 3.75957777928413158754001196342, 4.46120291315624812820933360632, 5.07538788930950666598416222140, 6.08529635580399366863320012564, 6.88297244069579931175325797636, 7.75596315170035840403601785916
