L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 6·11-s − 2·15-s + 8·17-s + 4·19-s − 21-s − 2·23-s − 25-s + 27-s + 10·29-s − 7·31-s + 6·33-s + 2·35-s + 2·37-s − 4·41-s + 7·43-s − 2·45-s − 6·47-s − 6·49-s + 8·51-s − 12·55-s + 4·57-s + 6·59-s − 5·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.516·15-s + 1.94·17-s + 0.917·19-s − 0.218·21-s − 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.25·31-s + 1.04·33-s + 0.338·35-s + 0.328·37-s − 0.624·41-s + 1.06·43-s − 0.298·45-s − 0.875·47-s − 6/7·49-s + 1.12·51-s − 1.61·55-s + 0.529·57-s + 0.781·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.616440035\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.616440035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.79190851781197979673446306843, −7.28484972630459505156903391971, −6.54321340500400660212501498861, −5.84927361172071044840732292050, −4.89833158388844958470721870732, −3.99387486918897381458182914710, −3.54525284154800107346081353018, −2.98082814400719713900365763428, −1.61485669148580227923442738929, −0.840334850409286993957797483059,
0.840334850409286993957797483059, 1.61485669148580227923442738929, 2.98082814400719713900365763428, 3.54525284154800107346081353018, 3.99387486918897381458182914710, 4.89833158388844958470721870732, 5.84927361172071044840732292050, 6.54321340500400660212501498861, 7.28484972630459505156903391971, 7.79190851781197979673446306843