| L(s) = 1 | + 3·5-s − 2·7-s + 4·13-s + 6·17-s − 8·19-s + 3·23-s + 4·25-s + 5·31-s − 6·35-s − 37-s + 10·43-s − 3·49-s + 6·53-s − 3·59-s + 4·61-s + 12·65-s − 67-s − 15·71-s + 4·73-s − 2·79-s + 6·83-s + 18·85-s + 9·89-s − 8·91-s − 24·95-s − 7·97-s + 18·101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s + 1.10·13-s + 1.45·17-s − 1.83·19-s + 0.625·23-s + 4/5·25-s + 0.898·31-s − 1.01·35-s − 0.164·37-s + 1.52·43-s − 3/7·49-s + 0.824·53-s − 0.390·59-s + 0.512·61-s + 1.48·65-s − 0.122·67-s − 1.78·71-s + 0.468·73-s − 0.225·79-s + 0.658·83-s + 1.95·85-s + 0.953·89-s − 0.838·91-s − 2.46·95-s − 0.710·97-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521276913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521276913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.585866889119459033715626592235, −7.63173684139913449353926662917, −6.66019819116288679958563294672, −6.08718240220327616520154502572, −5.73530160989426067882261609281, −4.68022649879253263009993418914, −3.70217639365433698551581447714, −2.86631612844463412939210052570, −1.95567073598448482222279023681, −0.926715754564195996924603331143,
0.926715754564195996924603331143, 1.95567073598448482222279023681, 2.86631612844463412939210052570, 3.70217639365433698551581447714, 4.68022649879253263009993418914, 5.73530160989426067882261609281, 6.08718240220327616520154502572, 6.66019819116288679958563294672, 7.63173684139913449353926662917, 8.585866889119459033715626592235
