| L(s) = 1 | − 3-s − 1.12·5-s + 7-s + 9-s + 4.76·11-s − 0.456·13-s + 1.12·15-s + 0.415·17-s − 7.63·19-s − 21-s + 1.58·23-s − 3.72·25-s − 27-s − 6.72·29-s + 5.89·31-s − 4.76·33-s − 1.12·35-s + 5.89·37-s + 0.456·39-s + 0.415·41-s − 9.43·43-s − 1.12·45-s − 11.2·47-s + 49-s − 0.415·51-s + 7.63·53-s − 5.38·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.504·5-s + 0.377·7-s + 0.333·9-s + 1.43·11-s − 0.126·13-s + 0.291·15-s + 0.100·17-s − 1.75·19-s − 0.218·21-s + 0.330·23-s − 0.745·25-s − 0.192·27-s − 1.24·29-s + 1.05·31-s − 0.829·33-s − 0.190·35-s + 0.969·37-s + 0.0730·39-s + 0.0648·41-s − 1.43·43-s − 0.168·45-s − 1.64·47-s + 0.142·49-s − 0.0581·51-s + 1.04·53-s − 0.725·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 1.12T + 5T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 0.456T + 13T^{2} \) |
| 17 | \( 1 - 0.415T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.72T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 0.415T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.934194216390723823961946029704, −6.89100920128260955732233698013, −6.49810328663136413581890010220, −5.74072774282036752932133933722, −4.75879424887899400465142556755, −4.17528097750483758162883178618, −3.53388752050284031465946627377, −2.18969267933053965763363448661, −1.29066046266251449697388093091, 0,
1.29066046266251449697388093091, 2.18969267933053965763363448661, 3.53388752050284031465946627377, 4.17528097750483758162883178618, 4.75879424887899400465142556755, 5.74072774282036752932133933722, 6.49810328663136413581890010220, 6.89100920128260955732233698013, 7.934194216390723823961946029704
