| L(s) = 1 | + 3-s + 5-s + 4.68·7-s + 9-s + 2.29·11-s − 4.97·13-s + 15-s − 2.97·17-s − 2.68·19-s + 4.68·21-s + 2.68·23-s + 25-s + 27-s + 2·29-s + 6.97·31-s + 2.29·33-s + 4.68·35-s − 4.39·37-s − 4.97·39-s + 11.3·41-s + 9.37·43-s + 45-s − 7.27·47-s + 14.9·49-s − 2.97·51-s + 2·53-s + 2.29·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.77·7-s + 0.333·9-s + 0.691·11-s − 1.38·13-s + 0.258·15-s − 0.722·17-s − 0.616·19-s + 1.02·21-s + 0.560·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 1.25·31-s + 0.399·33-s + 0.792·35-s − 0.722·37-s − 0.797·39-s + 1.77·41-s + 1.42·43-s + 0.149·45-s − 1.06·47-s + 2.13·49-s − 0.417·51-s + 0.274·53-s + 0.309·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.277021879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.277021879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 0.585T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 3.95T + 97T^{2} \) |
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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
−8.481182275456020529051879428097, −7.83834328087237765869672665770, −7.14926576789022755117576503508, −6.38042052374293757128324753759, −5.28770900701820343220553130160, −4.64250208422423109365138752872, −4.11262789093258421338317985499, −2.64401683932788002558649523917, −2.11046612306839665115103320443, −1.09748708760152999777154399496,
1.09748708760152999777154399496, 2.11046612306839665115103320443, 2.64401683932788002558649523917, 4.11262789093258421338317985499, 4.64250208422423109365138752872, 5.28770900701820343220553130160, 6.38042052374293757128324753759, 7.14926576789022755117576503508, 7.83834328087237765869672665770, 8.481182275456020529051879428097
