| L(s) = 1 | + 7·3-s + 34·7-s + 22·9-s + 27·11-s + 28·13-s + 21·17-s + 35·19-s + 238·21-s + 78·23-s − 35·27-s + 120·29-s − 182·31-s + 189·33-s − 146·37-s + 196·39-s + 357·41-s − 148·43-s + 84·47-s + 813·49-s + 147·51-s − 702·53-s + 245·57-s − 840·59-s + 238·61-s + 748·63-s + 461·67-s + 546·69-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 1.83·7-s + 0.814·9-s + 0.740·11-s + 0.597·13-s + 0.299·17-s + 0.422·19-s + 2.47·21-s + 0.707·23-s − 0.249·27-s + 0.768·29-s − 1.05·31-s + 0.996·33-s − 0.648·37-s + 0.804·39-s + 1.35·41-s − 0.524·43-s + 0.260·47-s + 2.37·49-s + 0.403·51-s − 1.81·53-s + 0.569·57-s − 1.85·59-s + 0.499·61-s + 1.49·63-s + 0.840·67-s + 0.952·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.336851446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.336851446\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 + 182 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 357 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 702 T + p^{3} T^{2} \) |
| 59 | \( 1 + 840 T + p^{3} T^{2} \) |
| 61 | \( 1 - 238 T + p^{3} T^{2} \) |
| 67 | \( 1 - 461 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 133 T + p^{3} T^{2} \) |
| 79 | \( 1 + 650 T + p^{3} T^{2} \) |
| 83 | \( 1 + 903 T + p^{3} T^{2} \) |
| 89 | \( 1 - 735 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1106 T + p^{3} T^{2} \) |
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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
−8.968954748806235099297005367809, −8.198876217286216483533647817498, −7.79327078191373108991289656777, −6.90294959805068661683219192317, −5.65822977025511638254009524695, −4.73435628700341201755362462571, −3.89876188250044964052991699108, −2.98604510631426408695606203645, −1.85965969134018873822036578716, −1.19822110011521449933990965202,
1.19822110011521449933990965202, 1.85965969134018873822036578716, 2.98604510631426408695606203645, 3.89876188250044964052991699108, 4.73435628700341201755362462571, 5.65822977025511638254009524695, 6.90294959805068661683219192317, 7.79327078191373108991289656777, 8.198876217286216483533647817498, 8.968954748806235099297005367809
