| L(s) = 1 | + 0.438·2-s − 2.16·3-s − 1.80·4-s − 1.47·5-s − 0.947·6-s − 4.80·7-s − 1.66·8-s + 1.67·9-s − 0.647·10-s + 3.90·12-s − 1.27·13-s − 2.10·14-s + 3.19·15-s + 2.88·16-s + 2.43·17-s + 0.735·18-s − 8.40·19-s + 2.67·20-s + 10.3·21-s − 5.24·23-s + 3.60·24-s − 2.81·25-s − 0.556·26-s + 2.86·27-s + 8.68·28-s − 0.804·29-s + 1.40·30-s + ⋯ |
| L(s) = 1 | + 0.309·2-s − 1.24·3-s − 0.903·4-s − 0.661·5-s − 0.386·6-s − 1.81·7-s − 0.590·8-s + 0.559·9-s − 0.204·10-s + 1.12·12-s − 0.352·13-s − 0.562·14-s + 0.825·15-s + 0.721·16-s + 0.591·17-s + 0.173·18-s − 1.92·19-s + 0.597·20-s + 2.26·21-s − 1.09·23-s + 0.736·24-s − 0.562·25-s − 0.109·26-s + 0.550·27-s + 1.64·28-s − 0.149·29-s + 0.255·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.438T + 2T^{2} \) |
| 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 8.40T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 + 0.804T + 29T^{2} \) |
| 31 | \( 1 + 0.544T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 1.59T + 97T^{2} \) |
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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
−7.43841943342368424964041250366, −6.54037852370644292053518422997, −6.04762418439172746135188407873, −5.64838240235819329286434960669, −4.63875196533285968399888692618, −4.01924076945658202549282683883, −3.48541009780493796277662312338, −2.42024010463391250969158792654, −0.60462042614728872514024752985, 0,
0.60462042614728872514024752985, 2.42024010463391250969158792654, 3.48541009780493796277662312338, 4.01924076945658202549282683883, 4.63875196533285968399888692618, 5.64838240235819329286434960669, 6.04762418439172746135188407873, 6.54037852370644292053518422997, 7.43841943342368424964041250366
