| L(s) = 1 | − 2.22·2-s + 1.19·3-s + 2.96·4-s + 2.18·5-s − 2.66·6-s + 7-s − 2.14·8-s − 1.56·9-s − 4.85·10-s − 5.84·11-s + 3.54·12-s − 4.49·13-s − 2.22·14-s + 2.60·15-s − 1.15·16-s + 4.28·17-s + 3.49·18-s + 2.48·19-s + 6.45·20-s + 1.19·21-s + 13.0·22-s − 2.32·23-s − 2.55·24-s − 0.244·25-s + 10.0·26-s − 5.46·27-s + 2.96·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.690·3-s + 1.48·4-s + 0.975·5-s − 1.08·6-s + 0.377·7-s − 0.756·8-s − 0.523·9-s − 1.53·10-s − 1.76·11-s + 1.02·12-s − 1.24·13-s − 0.595·14-s + 0.673·15-s − 0.288·16-s + 1.03·17-s + 0.824·18-s + 0.569·19-s + 1.44·20-s + 0.260·21-s + 2.77·22-s − 0.484·23-s − 0.522·24-s − 0.0489·25-s + 1.96·26-s − 1.05·27-s + 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9941708742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9941708742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 - 3.08T + 59T^{2} \) |
| 61 | \( 1 + 6.77T + 61T^{2} \) |
| 67 | \( 1 + 5.38T + 67T^{2} \) |
| 71 | \( 1 + 3.33T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + 0.269T + 97T^{2} \) |
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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
−8.073397327749517891691133643578, −7.58132785170320627232423410654, −7.27568920365628758892933125646, −5.82758181667336765613522350384, −5.54898742927996365745797561169, −4.54539780634927398337950124555, −3.04102779306834565627009755919, −2.44772351580190349376613461649, −1.92354440777452828808265463638, −0.61441284947739139612762379858,
0.61441284947739139612762379858, 1.92354440777452828808265463638, 2.44772351580190349376613461649, 3.04102779306834565627009755919, 4.54539780634927398337950124555, 5.54898742927996365745797561169, 5.82758181667336765613522350384, 7.27568920365628758892933125646, 7.58132785170320627232423410654, 8.073397327749517891691133643578
