| L(s) = 1 | + 3-s − 0.978·5-s − 4.58·7-s + 9-s − 0.922·11-s − 13-s − 0.978·15-s + 1.38·17-s + 5.16·19-s − 4.58·21-s + 1.65·23-s − 4.04·25-s + 27-s − 0.432·29-s + 3.41·31-s − 0.922·33-s + 4.48·35-s − 0.217·37-s − 39-s + 10.4·41-s − 2.25·43-s − 0.978·45-s + 2.75·47-s + 14.0·49-s + 1.38·51-s − 2.95·53-s + 0.901·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.437·5-s − 1.73·7-s + 0.333·9-s − 0.277·11-s − 0.277·13-s − 0.252·15-s + 0.335·17-s + 1.18·19-s − 1.00·21-s + 0.346·23-s − 0.808·25-s + 0.192·27-s − 0.0802·29-s + 0.613·31-s − 0.160·33-s + 0.758·35-s − 0.0357·37-s − 0.160·39-s + 1.62·41-s − 0.343·43-s − 0.145·45-s + 0.401·47-s + 2.00·49-s + 0.193·51-s − 0.405·53-s + 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 0.978T + 5T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 11 | \( 1 + 0.922T + 11T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 + 0.432T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 0.217T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 + 9.34T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−7.35341365039842918342801003930, −6.77787307595261720791432376249, −5.98494267963417872457645960410, −5.37247739946338232731378748985, −4.31598630451899511153697682244, −3.66875536029132475565321879420, −3.01231612372289675360041194326, −2.50291726180055559421691932405, −1.10901356625750322198916765275, 0,
1.10901356625750322198916765275, 2.50291726180055559421691932405, 3.01231612372289675360041194326, 3.66875536029132475565321879420, 4.31598630451899511153697682244, 5.37247739946338232731378748985, 5.98494267963417872457645960410, 6.77787307595261720791432376249, 7.35341365039842918342801003930
