| L(s) = 1 | + 3-s + 3.18·5-s − 1.39·7-s + 9-s − 2.54·11-s + 5.18·13-s + 3.18·15-s − 2.90·17-s + 6.55·19-s − 1.39·21-s + 2.68·23-s + 5.12·25-s + 27-s − 3.50·29-s + 6.85·31-s − 2.54·33-s − 4.43·35-s + 4.23·37-s + 5.18·39-s − 1.88·41-s − 6.49·43-s + 3.18·45-s − 7.80·47-s − 5.05·49-s − 2.90·51-s + 10.5·53-s − 8.10·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.42·5-s − 0.526·7-s + 0.333·9-s − 0.767·11-s + 1.43·13-s + 0.821·15-s − 0.704·17-s + 1.50·19-s − 0.304·21-s + 0.559·23-s + 1.02·25-s + 0.192·27-s − 0.650·29-s + 1.23·31-s − 0.443·33-s − 0.749·35-s + 0.696·37-s + 0.829·39-s − 0.295·41-s − 0.990·43-s + 0.474·45-s − 1.13·47-s − 0.722·49-s − 0.406·51-s + 1.44·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.219138814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.219138814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 1.88T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 0.200T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 1.95T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−8.580655793520750278281327612854, −7.81571638520948254391853435499, −6.83770057994901974598981232940, −6.26648430234469682708989258993, −5.53848941456840287071629894143, −4.82077786775081081478794438691, −3.58558592999184362212304912803, −2.92078746362485779786845212729, −2.04789893862492663659179353398, −1.06241690942521870357835016880,
1.06241690942521870357835016880, 2.04789893862492663659179353398, 2.92078746362485779786845212729, 3.58558592999184362212304912803, 4.82077786775081081478794438691, 5.53848941456840287071629894143, 6.26648430234469682708989258993, 6.83770057994901974598981232940, 7.81571638520948254391853435499, 8.580655793520750278281327612854
