| L(s) = 1 | − 2.77·3-s + 5-s + 4.55·7-s + 4.68·9-s − 2.90·13-s − 2.77·15-s + 7.08·17-s + 7.14·19-s − 12.6·21-s − 1.86·23-s + 25-s − 4.66·27-s + 1.01·29-s + 4.41·31-s + 4.55·35-s − 5.39·37-s + 8.03·39-s + 4.48·41-s + 6.73·43-s + 4.68·45-s + 7.85·47-s + 13.7·49-s − 19.6·51-s + 0.714·53-s − 19.8·57-s − 3.80·59-s − 2.11·61-s + ⋯ |
| L(s) = 1 | − 1.60·3-s + 0.447·5-s + 1.72·7-s + 1.56·9-s − 0.804·13-s − 0.715·15-s + 1.71·17-s + 1.63·19-s − 2.75·21-s − 0.389·23-s + 0.200·25-s − 0.898·27-s + 0.187·29-s + 0.792·31-s + 0.769·35-s − 0.886·37-s + 1.28·39-s + 0.700·41-s + 1.02·43-s + 0.698·45-s + 1.14·47-s + 1.96·49-s − 2.75·51-s + 0.0981·53-s − 2.62·57-s − 0.495·59-s − 0.270·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736277247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736277247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 - 7.08T + 17T^{2} \) |
| 19 | \( 1 - 7.14T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 0.714T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.83233152504830794141708606654, −7.68712093060732694807717594959, −6.77935305491454215013959707644, −5.73569169591635025913740313807, −5.40312650629579881113890482450, −4.92407455775630839864910905210, −4.10120553610989676928462608786, −2.74382616641230408853955889426, −1.49058152752262686429228220669, −0.899129935852657515928910451153,
0.899129935852657515928910451153, 1.49058152752262686429228220669, 2.74382616641230408853955889426, 4.10120553610989676928462608786, 4.92407455775630839864910905210, 5.40312650629579881113890482450, 5.73569169591635025913740313807, 6.77935305491454215013959707644, 7.68712093060732694807717594959, 7.83233152504830794141708606654
