| L(s) = 1 | − 2.20·3-s + 1.56·7-s + 1.86·9-s − 5.47·11-s − 4.33·13-s + 4.06·17-s − 7.14·19-s − 3.44·21-s − 1.82·23-s + 2.50·27-s + 29-s + 4.61·31-s + 12.0·33-s + 7.54·37-s + 9.55·39-s − 6.52·41-s + 4.02·43-s − 6.44·47-s − 4.55·49-s − 8.97·51-s + 7.35·53-s + 15.7·57-s + 8.82·59-s − 8.19·61-s + 2.91·63-s − 10.0·67-s + 4.03·69-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 0.590·7-s + 0.621·9-s − 1.64·11-s − 1.20·13-s + 0.986·17-s − 1.63·19-s − 0.752·21-s − 0.381·23-s + 0.482·27-s + 0.185·29-s + 0.828·31-s + 2.09·33-s + 1.24·37-s + 1.53·39-s − 1.01·41-s + 0.613·43-s − 0.940·47-s − 0.651·49-s − 1.25·51-s + 1.00·53-s + 2.08·57-s + 1.14·59-s − 1.04·61-s + 0.366·63-s − 1.22·67-s + 0.485·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6505128524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6505128524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 + 7.14T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 - 4.02T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 7.35T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 0.416T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.558676783291030709701390792299, −7.938368941356261096915352718862, −7.28449664142933631951096051331, −6.28672477038524939886148857823, −5.66880284548583327292823454798, −4.88787731030777140590621541544, −4.51329314534814232166014145420, −2.96045062337754962407177337513, −2.04597440030941806190985582453, −0.50497651562037122282609479887,
0.50497651562037122282609479887, 2.04597440030941806190985582453, 2.96045062337754962407177337513, 4.51329314534814232166014145420, 4.88787731030777140590621541544, 5.66880284548583327292823454798, 6.28672477038524939886148857823, 7.28449664142933631951096051331, 7.938368941356261096915352718862, 8.558676783291030709701390792299
