| L(s) = 1 | − 2.35·3-s + 2.85·5-s − 0.141·7-s + 2.54·9-s + 11-s + 5.42·13-s − 6.73·15-s + 2·17-s − 19-s + 0.332·21-s + 1.40·23-s + 3.17·25-s + 1.07·27-s − 5.80·29-s + 10.7·31-s − 2.35·33-s − 0.403·35-s + 0.313·37-s − 12.7·39-s + 8.33·41-s − 4.16·43-s + 7.27·45-s − 1.09·47-s − 6.98·49-s − 4.70·51-s + 2·53-s + 2.85·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 1.27·5-s − 0.0533·7-s + 0.848·9-s + 0.301·11-s + 1.50·13-s − 1.73·15-s + 0.485·17-s − 0.229·19-s + 0.0725·21-s + 0.292·23-s + 0.634·25-s + 0.206·27-s − 1.07·29-s + 1.93·31-s − 0.409·33-s − 0.0682·35-s + 0.0515·37-s − 2.04·39-s + 1.30·41-s − 0.635·43-s + 1.08·45-s − 0.159·47-s − 0.997·49-s − 0.659·51-s + 0.274·53-s + 0.385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656331310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656331310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 + 0.141T + 7T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 0.313T + 37T^{2} \) |
| 41 | \( 1 - 8.33T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 - 0.969T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 1.71T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 7.87T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.763663708171398518814294503013, −7.84518506016956208489857566884, −6.68762302259520821099564629397, −6.21035345860609320730054592944, −5.79324226790867327131802315693, −5.06075910948859889058815137531, −4.14194120348923618317526728211, −3.00305953409890745961770835555, −1.71764616959172982873608564254, −0.881377464071613570756135362286,
0.881377464071613570756135362286, 1.71764616959172982873608564254, 3.00305953409890745961770835555, 4.14194120348923618317526728211, 5.06075910948859889058815137531, 5.79324226790867327131802315693, 6.21035345860609320730054592944, 6.68762302259520821099564629397, 7.84518506016956208489857566884, 8.763663708171398518814294503013
