| L(s) = 1 | − 2-s − 1.37·3-s + 4-s + 5-s + 1.37·6-s − 1.05·7-s − 8-s − 1.10·9-s − 10-s − 11-s − 1.37·12-s + 3.04·13-s + 1.05·14-s − 1.37·15-s + 16-s − 0.292·17-s + 1.10·18-s − 1.01·19-s + 20-s + 1.45·21-s + 22-s − 3.24·23-s + 1.37·24-s + 25-s − 3.04·26-s + 5.65·27-s − 1.05·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.794·3-s + 0.5·4-s + 0.447·5-s + 0.561·6-s − 0.398·7-s − 0.353·8-s − 0.368·9-s − 0.316·10-s − 0.301·11-s − 0.397·12-s + 0.844·13-s + 0.281·14-s − 0.355·15-s + 0.250·16-s − 0.0709·17-s + 0.260·18-s − 0.233·19-s + 0.223·20-s + 0.316·21-s + 0.213·22-s − 0.676·23-s + 0.280·24-s + 0.200·25-s − 0.596·26-s + 1.08·27-s − 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 1.37T + 3T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 0.292T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 7.64T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 + 9.53T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 - 1.37T + 71T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 8.43T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.52745399306268263543612080922, −6.56176066018355851364090188098, −6.25084678072722914919755356049, −5.61586290080777337458311475160, −4.88622316601342549042309947825, −3.82914139372039992398694033325, −2.94839424537467798328272200013, −2.08679002452303440646412663170, −1.02501916762221506733718092335, 0,
1.02501916762221506733718092335, 2.08679002452303440646412663170, 2.94839424537467798328272200013, 3.82914139372039992398694033325, 4.88622316601342549042309947825, 5.61586290080777337458311475160, 6.25084678072722914919755356049, 6.56176066018355851364090188098, 7.52745399306268263543612080922
