| L(s) = 1 | + 4·5-s + 7-s − 3·9-s + 2·13-s + 4·17-s − 6·19-s + 4·23-s + 11·25-s − 2·29-s − 2·31-s + 4·35-s − 10·37-s − 4·41-s − 8·43-s − 12·45-s + 2·47-s + 49-s − 6·53-s + 12·59-s − 14·61-s − 3·63-s + 8·65-s + 12·67-s − 8·71-s − 4·73-s + 9·81-s − 6·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 9-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 0.676·35-s − 1.64·37-s − 0.624·41-s − 1.21·43-s − 1.78·45-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 1.79·61-s − 0.377·63-s + 0.992·65-s + 1.46·67-s − 0.949·71-s − 0.468·73-s + 81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.56215553989019, −14.21861043723439, −13.62392570951131, −13.32123515115192, −12.73817381366897, −12.23225506371848, −11.55813478927210, −10.94738385272255, −10.56577576855878, −10.12271848711037, −9.467823681599584, −9.006446201198212, −8.446945943349398, −8.202683066477130, −7.074626520015256, −6.748817343242994, −5.987507861969538, −5.700825557510661, −5.177876709222920, −4.646536070955191, −3.588056659497829, −3.090317603803582, −2.336370442214266, −1.760734372756468, −1.211592819996238, 0,
1.211592819996238, 1.760734372756468, 2.336370442214266, 3.090317603803582, 3.588056659497829, 4.646536070955191, 5.177876709222920, 5.700825557510661, 5.987507861969538, 6.748817343242994, 7.074626520015256, 8.202683066477130, 8.446945943349398, 9.006446201198212, 9.467823681599584, 10.12271848711037, 10.56577576855878, 10.94738385272255, 11.55813478927210, 12.23225506371848, 12.73817381366897, 13.32123515115192, 13.62392570951131, 14.21861043723439, 14.56215553989019
