| L(s) = 1 | + 1.06·3-s − 2.95·7-s − 1.85·9-s − 5.89·11-s + 3.64·13-s − 4.68·17-s − 5.73·19-s − 3.15·21-s + 23-s − 5.19·27-s + 10.3·29-s − 7.29·31-s − 6.30·33-s + 0.612·37-s + 3.90·39-s − 9.10·41-s + 1.58·43-s + 6.22·47-s + 1.71·49-s − 5.01·51-s − 3.17·53-s − 6.13·57-s − 1.38·59-s + 6.02·61-s + 5.48·63-s + 13.3·67-s + 1.06·69-s + ⋯ |
| L(s) = 1 | + 0.617·3-s − 1.11·7-s − 0.618·9-s − 1.77·11-s + 1.01·13-s − 1.13·17-s − 1.31·19-s − 0.689·21-s + 0.208·23-s − 0.999·27-s + 1.92·29-s − 1.31·31-s − 1.09·33-s + 0.100·37-s + 0.624·39-s − 1.42·41-s + 0.241·43-s + 0.908·47-s + 0.245·49-s − 0.701·51-s − 0.436·53-s − 0.812·57-s − 0.180·59-s + 0.770·61-s + 0.690·63-s + 1.63·67-s + 0.128·69-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
| \(L(1)\) |
\(\approx\) |
\(0.9484005233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9484005233\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.06T + 3T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 0.612T + 37T^{2} \) |
| 41 | \( 1 + 9.10T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 6.02T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 6.22T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 - 7.01T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−8.009510973303182089223690144073, −6.86162495542377419096780503291, −6.49973145754636468503167309347, −5.69642036099550339067315134476, −5.01109553284090961088743370265, −4.04429541656510442479886350228, −3.31660668677406307122945461076, −2.64706030773073271341179271253, −2.08571101752715951295694674557, −0.41705501749032012545979855092,
0.41705501749032012545979855092, 2.08571101752715951295694674557, 2.64706030773073271341179271253, 3.31660668677406307122945461076, 4.04429541656510442479886350228, 5.01109553284090961088743370265, 5.69642036099550339067315134476, 6.49973145754636468503167309347, 6.86162495542377419096780503291, 8.009510973303182089223690144073
