| L(s) = 1 | + 3.04·5-s − 1.31·7-s − 3.04·13-s + 1.27·17-s + 1.31·19-s + 4.27·25-s − 1.27·29-s − 2.27·31-s − 4·35-s − 9.54·37-s − 5.27·41-s − 9.55·43-s + 5.25·47-s − 5.27·49-s − 9.97·53-s + 12.1·59-s − 1.31·61-s − 9.27·65-s + 14.8·67-s + 6.92·71-s − 8.24·73-s − 13.4·79-s − 12·83-s + 3.88·85-s − 3.04·89-s + 4·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 1.36·5-s − 0.496·7-s − 0.844·13-s + 0.309·17-s + 0.301·19-s + 0.854·25-s − 0.236·29-s − 0.408·31-s − 0.676·35-s − 1.56·37-s − 0.823·41-s − 1.45·43-s + 0.766·47-s − 0.753·49-s − 1.36·53-s + 1.58·59-s − 0.168·61-s − 1.15·65-s + 1.81·67-s + 0.822·71-s − 0.964·73-s − 1.51·79-s − 1.31·83-s + 0.421·85-s − 0.322·89-s + 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 + 2.27T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 + 9.55T + 43T^{2} \) |
| 47 | \( 1 - 5.25T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 + 0.450T + 97T^{2} \) |
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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
−7.17018355248199266304263992123, −6.79204913306258123390981599731, −6.00066545172095956564428771860, −5.34694595636969487311644445064, −4.90620747371062984824349446969, −3.72501344419608991655930647117, −2.99574133066799204968578937057, −2.14341255241724603461361463389, −1.44728548892036945212803827934, 0,
1.44728548892036945212803827934, 2.14341255241724603461361463389, 2.99574133066799204968578937057, 3.72501344419608991655930647117, 4.90620747371062984824349446969, 5.34694595636969487311644445064, 6.00066545172095956564428771860, 6.79204913306258123390981599731, 7.17018355248199266304263992123
