| L(s) = 1 | − 3·3-s + 3·5-s + 2·7-s + 6·9-s − 11-s − 9·15-s − 6·17-s + 4·19-s − 6·21-s − 23-s + 4·25-s − 9·27-s + 8·29-s + 7·31-s + 3·33-s + 6·35-s + 37-s + 4·41-s + 6·43-s + 18·45-s + 8·47-s − 3·49-s + 18·51-s − 2·53-s − 3·55-s − 12·57-s − 59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 0.755·7-s + 2·9-s − 0.301·11-s − 2.32·15-s − 1.45·17-s + 0.917·19-s − 1.30·21-s − 0.208·23-s + 4/5·25-s − 1.73·27-s + 1.48·29-s + 1.25·31-s + 0.522·33-s + 1.01·35-s + 0.164·37-s + 0.624·41-s + 0.914·43-s + 2.68·45-s + 1.16·47-s − 3/7·49-s + 2.52·51-s − 0.274·53-s − 0.404·55-s − 1.58·57-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170561290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170561290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.53066595506174115405880345862, −9.904335594569403495563534809786, −8.915095754172688127714174793157, −7.61244870210294828756700835228, −6.49345737138044489838835204705, −6.02593204701964171723236311732, −5.07494670300081394122416170846, −4.52202018274487321804857018627, −2.35704636913925935503557402485, −1.05793233766724449207828741370,
1.05793233766724449207828741370, 2.35704636913925935503557402485, 4.52202018274487321804857018627, 5.07494670300081394122416170846, 6.02593204701964171723236311732, 6.49345737138044489838835204705, 7.61244870210294828756700835228, 8.915095754172688127714174793157, 9.904335594569403495563534809786, 10.53066595506174115405880345862
