L(s) = 1 | + 1.21·3-s + 5-s + 0.325·7-s − 1.51·9-s + 4.76·11-s − 1.01·13-s + 1.21·15-s − 4.24·17-s + 8.57·19-s + 0.396·21-s + 5.14·23-s + 25-s − 5.50·27-s − 8.53·29-s − 4.46·31-s + 5.80·33-s + 0.325·35-s + 2.83·37-s − 1.23·39-s + 12.2·41-s + 4.77·43-s − 1.51·45-s + 10.0·47-s − 6.89·49-s − 5.18·51-s − 12.0·53-s + 4.76·55-s + ⋯ |
L(s) = 1 | + 0.704·3-s + 0.447·5-s + 0.122·7-s − 0.503·9-s + 1.43·11-s − 0.281·13-s + 0.314·15-s − 1.03·17-s + 1.96·19-s + 0.0865·21-s + 1.07·23-s + 0.200·25-s − 1.05·27-s − 1.58·29-s − 0.801·31-s + 1.01·33-s + 0.0549·35-s + 0.466·37-s − 0.198·39-s + 1.91·41-s + 0.728·43-s − 0.225·45-s + 1.46·47-s − 0.984·49-s − 0.725·51-s − 1.65·53-s + 0.642·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.083419074\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.083419074\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 1.21T + 3T^{2} \) |
| 7 | \( 1 - 0.325T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 8.57T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 8.53T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.53T + 73T^{2} \) |
| 79 | \( 1 + 9.60T + 79T^{2} \) |
| 83 | \( 1 + 4.05T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.100872726962047726854612401595, −7.32806409565458366784439834184, −6.83954730933848499325803706220, −5.81840623975994793496122826769, −5.36651866668504038100022142565, −4.27754733924927199317627440190, −3.58000841000611171728802200737, −2.77052128464324801268999771973, −1.95066112054012415141695097924, −0.921285163443610006465164496903,
0.921285163443610006465164496903, 1.95066112054012415141695097924, 2.77052128464324801268999771973, 3.58000841000611171728802200737, 4.27754733924927199317627440190, 5.36651866668504038100022142565, 5.81840623975994793496122826769, 6.83954730933848499325803706220, 7.32806409565458366784439834184, 8.100872726962047726854612401595