| L(s) = 1 | − 1.89·2-s − 2.41·3-s + 1.58·4-s − 0.628·5-s + 4.57·6-s + 4.16·7-s + 0.794·8-s + 2.84·9-s + 1.18·10-s − 3.82·12-s + 0.733·13-s − 7.88·14-s + 1.52·15-s − 4.66·16-s + 5.31·17-s − 5.38·18-s + 0.379·19-s − 0.993·20-s − 10.0·21-s − 6.92·23-s − 1.92·24-s − 4.60·25-s − 1.38·26-s + 0.374·27-s + 6.58·28-s + 4.23·29-s − 2.87·30-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 1.39·3-s + 0.790·4-s − 0.281·5-s + 1.86·6-s + 1.57·7-s + 0.280·8-s + 0.948·9-s + 0.376·10-s − 1.10·12-s + 0.203·13-s − 2.10·14-s + 0.392·15-s − 1.16·16-s + 1.28·17-s − 1.26·18-s + 0.0870·19-s − 0.222·20-s − 2.19·21-s − 1.44·23-s − 0.392·24-s − 0.920·25-s − 0.272·26-s + 0.0720·27-s + 1.24·28-s + 0.787·29-s − 0.525·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5563782532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5563782532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 0.628T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 13 | \( 1 - 0.733T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 0.379T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 5.50T + 31T^{2} \) |
| 37 | \( 1 + 4.93T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 + 1.82T + 59T^{2} \) |
| 67 | \( 1 - 5.21T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 4.61T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.88038167984888953516371794940, −7.53153772608214203343326109285, −6.63538903450991706699722028393, −5.77939552357156047446521467111, −5.22750974456273548134402847093, −4.53757713399654543151906877130, −3.72473613410635958301783732992, −2.09627623982424063627662530076, −1.39085049197057639834759847033, −0.55298558100684443047921191278,
0.55298558100684443047921191278, 1.39085049197057639834759847033, 2.09627623982424063627662530076, 3.72473613410635958301783732992, 4.53757713399654543151906877130, 5.22750974456273548134402847093, 5.77939552357156047446521467111, 6.63538903450991706699722028393, 7.53153772608214203343326109285, 7.88038167984888953516371794940
