| L(s) = 1 | − 2.82·3-s − 1.41·5-s + 2.82·7-s + 5.00·9-s − 2.82·11-s − 4·13-s + 4.00·15-s − 4·19-s − 8.00·21-s + 2.82·23-s − 2.99·25-s − 5.65·27-s + 7.07·29-s + 8.48·31-s + 8.00·33-s − 4.00·35-s + 7.07·37-s + 11.3·39-s − 9.89·41-s − 4·43-s − 7.07·45-s + 8·47-s + 1.00·49-s − 4·53-s + 4.00·55-s + 11.3·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 1.63·3-s − 0.632·5-s + 1.06·7-s + 1.66·9-s − 0.852·11-s − 1.10·13-s + 1.03·15-s − 0.917·19-s − 1.74·21-s + 0.589·23-s − 0.599·25-s − 1.08·27-s + 1.31·29-s + 1.52·31-s + 1.39·33-s − 0.676·35-s + 1.16·37-s + 1.81·39-s − 1.54·41-s − 0.609·43-s − 1.05·45-s + 1.16·47-s + 0.142·49-s − 0.549·53-s + 0.539·55-s + 1.49·57-s − 1.56·59-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 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.24560348156517794653723378490, −6.67831737454573541212399155481, −5.94665872389786087644022428409, −5.17583760536786989542591593900, −4.63700732292032321125566644207, −4.40165257091735271347889439811, −2.99910442907357089664403704108, −2.03568425776735406635797278960, −0.908338969570871756968806664416, 0,
0.908338969570871756968806664416, 2.03568425776735406635797278960, 2.99910442907357089664403704108, 4.40165257091735271347889439811, 4.63700732292032321125566644207, 5.17583760536786989542591593900, 5.94665872389786087644022428409, 6.67831737454573541212399155481, 7.24560348156517794653723378490
