L(s) = 1 | − 0.523·3-s + 0.476·7-s − 2.72·9-s + 1.67·11-s − 2.67·13-s − 1.67·17-s − 7.92·19-s − 0.249·21-s + 23-s + 3·27-s − 2.20·29-s − 6.77·31-s − 0.878·33-s − 4·37-s + 1.40·39-s + 5.97·41-s − 0.402·43-s + 1.79·47-s − 6.77·49-s + 0.878·51-s + 10.8·53-s + 4.15·57-s − 9.45·59-s + 6.32·61-s − 1.29·63-s + 14.7·67-s − 0.523·69-s + ⋯ |
L(s) = 1 | − 0.302·3-s + 0.179·7-s − 0.908·9-s + 0.505·11-s − 0.742·13-s − 0.406·17-s − 1.81·19-s − 0.0544·21-s + 0.208·23-s + 0.577·27-s − 0.408·29-s − 1.21·31-s − 0.153·33-s − 0.657·37-s + 0.224·39-s + 0.933·41-s − 0.0614·43-s + 0.262·47-s − 0.967·49-s + 0.123·51-s + 1.48·53-s + 0.550·57-s − 1.23·59-s + 0.809·61-s − 0.163·63-s + 1.79·67-s − 0.0630·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9859202668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9859202668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 0.523T + 3T^{2} \) |
| 7 | \( 1 - 0.476T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 0.402T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 - 6.08T + 97T^{2} \) |
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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
−7.69649660755306145535543418347, −6.94617859261127205119841310913, −6.36104343221317024531291609743, −5.68060810971064131737565793414, −4.99814279989091926099130650780, −4.26121654885112017540800539976, −3.51867201159622021550514548458, −2.48278230264313319292850298447, −1.87506498338587031880748679238, −0.46237454836313022464386416365,
0.46237454836313022464386416365, 1.87506498338587031880748679238, 2.48278230264313319292850298447, 3.51867201159622021550514548458, 4.26121654885112017540800539976, 4.99814279989091926099130650780, 5.68060810971064131737565793414, 6.36104343221317024531291609743, 6.94617859261127205119841310913, 7.69649660755306145535543418347