| 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
| \(L(1)\) |
\(\approx\) |
\(0.6222693756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6222693756\) |
| \(L(\frac{3}{2})\) |
|
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
−8.630390342237459692406257615233, −7.38972203634515397626425043578, −7.06277198156207755689974452437, −6.20592562068817042967307285068, −5.60713997679657661864174451610, −4.87100626249852007035624277326, −3.83977398129389436157749585977, −2.89107152275067715985896925480, −2.14639701933083279396015740907, −0.43721770791480943591712467496,
0.43721770791480943591712467496, 2.14639701933083279396015740907, 2.89107152275067715985896925480, 3.83977398129389436157749585977, 4.87100626249852007035624277326, 5.60713997679657661864174451610, 6.20592562068817042967307285068, 7.06277198156207755689974452437, 7.38972203634515397626425043578, 8.630390342237459692406257615233
