| L(s) = 1 | − 2-s + 0.764·3-s + 4-s − 1.75·5-s − 0.764·6-s − 8-s − 2.41·9-s + 1.75·10-s − 6.17·11-s + 0.764·12-s − 4.95·13-s − 1.34·15-s + 16-s − 0.151·17-s + 2.41·18-s − 6.10·19-s − 1.75·20-s + 6.17·22-s + 7.24·23-s − 0.764·24-s − 1.90·25-s + 4.95·26-s − 4.14·27-s − 8.89·29-s + 1.34·30-s − 7.29·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.441·3-s + 0.5·4-s − 0.786·5-s − 0.312·6-s − 0.353·8-s − 0.805·9-s + 0.556·10-s − 1.86·11-s + 0.220·12-s − 1.37·13-s − 0.347·15-s + 0.250·16-s − 0.0368·17-s + 0.569·18-s − 1.40·19-s − 0.393·20-s + 1.31·22-s + 1.51·23-s − 0.156·24-s − 0.381·25-s + 0.971·26-s − 0.796·27-s − 1.65·29-s + 0.245·30-s − 1.31·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03703784223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03703784223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 0.764T + 3T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 + 0.151T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 2.95T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 + 9.57T + 97T^{2} \) |
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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
−7.950927492450514690794795024164, −7.47412186588234826606017554740, −6.79108255787384990002423235004, −5.67539929330416374228580116236, −5.17360286526773653098989671643, −4.26723374988508787890227356373, −3.22455248446108894096058124030, −2.62863260648976109787450797797, −1.96350953982310482708620748691, −0.094808776643860602047132880038,
0.094808776643860602047132880038, 1.96350953982310482708620748691, 2.62863260648976109787450797797, 3.22455248446108894096058124030, 4.26723374988508787890227356373, 5.17360286526773653098989671643, 5.67539929330416374228580116236, 6.79108255787384990002423235004, 7.47412186588234826606017554740, 7.950927492450514690794795024164
