L(s) = 1 | + 1.39·2-s + 3-s − 0.0409·4-s + 5-s + 1.39·6-s + 4.26·7-s − 2.85·8-s + 9-s + 1.39·10-s − 0.448·11-s − 0.0409·12-s − 3.19·13-s + 5.97·14-s + 15-s − 3.91·16-s − 3.39·17-s + 1.39·18-s + 3.10·19-s − 0.0409·20-s + 4.26·21-s − 0.628·22-s + 1.79·23-s − 2.85·24-s + 25-s − 4.46·26-s + 27-s − 0.174·28-s + ⋯ |
L(s) = 1 | + 0.989·2-s + 0.577·3-s − 0.0204·4-s + 0.447·5-s + 0.571·6-s + 1.61·7-s − 1.00·8-s + 0.333·9-s + 0.442·10-s − 0.135·11-s − 0.0118·12-s − 0.884·13-s + 1.59·14-s + 0.258·15-s − 0.979·16-s − 0.822·17-s + 0.329·18-s + 0.712·19-s − 0.00915·20-s + 0.931·21-s − 0.133·22-s + 0.373·23-s − 0.583·24-s + 0.200·25-s − 0.875·26-s + 0.192·27-s − 0.0330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.738552372\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.738552372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 11 | \( 1 + 0.448T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 41 | \( 1 + 6.01T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.239660065900289706808097319030, −7.26204179361502582483894060432, −6.70242312883402571023053657287, −5.46338128858320593981051101695, −5.22106425189933663512682999873, −4.45641131280342732639096721837, −3.89492523143274038146905585270, −2.66653826712071516702300775817, −2.27097045580286282889368550027, −0.998423185854786288951510670253,
0.998423185854786288951510670253, 2.27097045580286282889368550027, 2.66653826712071516702300775817, 3.89492523143274038146905585270, 4.45641131280342732639096721837, 5.22106425189933663512682999873, 5.46338128858320593981051101695, 6.70242312883402571023053657287, 7.26204179361502582483894060432, 8.239660065900289706808097319030