L(s) = 1 | + 1.77·2-s + 3-s + 1.14·4-s − 5-s + 1.77·6-s + 2.10·7-s − 1.52·8-s + 9-s − 1.77·10-s − 3.13·11-s + 1.14·12-s − 13-s + 3.72·14-s − 15-s − 4.97·16-s + 1.09·17-s + 1.77·18-s + 7.18·19-s − 1.14·20-s + 2.10·21-s − 5.55·22-s + 5.25·23-s − 1.52·24-s + 25-s − 1.77·26-s + 27-s + 2.40·28-s + ⋯ |
L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.571·4-s − 0.447·5-s + 0.723·6-s + 0.794·7-s − 0.537·8-s + 0.333·9-s − 0.560·10-s − 0.945·11-s + 0.329·12-s − 0.277·13-s + 0.995·14-s − 0.258·15-s − 1.24·16-s + 0.264·17-s + 0.417·18-s + 1.64·19-s − 0.255·20-s + 0.458·21-s − 1.18·22-s + 1.09·23-s − 0.310·24-s + 0.200·25-s − 0.347·26-s + 0.192·27-s + 0.453·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.510529359\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.510529359\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + 3.13T + 11T^{2} \) |
| 17 | \( 1 - 1.09T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 37 | \( 1 + 0.201T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 2.33T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 + 5.76T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−8.024459132953252876093422538252, −7.32982220529933237729759217233, −6.67621621177401495459541683998, −5.56315694517663607128119495999, −5.00202233971895514629244292954, −4.63080747663566881765915216206, −3.55952643900934723872738854637, −3.05865978843428311695657506034, −2.28199113367014569118072457087, −0.912813124773666594534352571436,
0.912813124773666594534352571436, 2.28199113367014569118072457087, 3.05865978843428311695657506034, 3.55952643900934723872738854637, 4.63080747663566881765915216206, 5.00202233971895514629244292954, 5.56315694517663607128119495999, 6.67621621177401495459541683998, 7.32982220529933237729759217233, 8.024459132953252876093422538252