| L(s) = 1 | + 2.53·2-s + 4.41·4-s − 5-s + 2.41·7-s + 6.10·8-s − 2.53·10-s − 1.65·11-s + 13-s + 6.10·14-s + 6.63·16-s + 4.29·17-s − 3.22·19-s − 4.41·20-s − 4.18·22-s + 0.305·23-s + 25-s + 2.53·26-s + 10.6·28-s + 3.71·29-s + 0.453·31-s + 4.59·32-s + 10.8·34-s − 2.41·35-s + 5.26·37-s − 8.17·38-s − 6.10·40-s + 12.3·41-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s − 0.447·5-s + 0.911·7-s + 2.15·8-s − 0.800·10-s − 0.498·11-s + 0.277·13-s + 1.63·14-s + 1.65·16-s + 1.04·17-s − 0.740·19-s − 0.986·20-s − 0.892·22-s + 0.0636·23-s + 0.200·25-s + 0.496·26-s + 2.01·28-s + 0.690·29-s + 0.0814·31-s + 0.812·32-s + 1.86·34-s − 0.407·35-s + 0.866·37-s − 1.32·38-s − 0.965·40-s + 1.92·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.694542793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.694542793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 - 0.305T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 - 0.453T + 31T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 - 8.83T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 2.04T + 73T^{2} \) |
| 79 | \( 1 + 0.319T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.84946360791647026742697521895, −7.44208403537851787648727669219, −6.49720552104566594219101665364, −5.78833706020384958605153062717, −5.22689540095913343017114111822, −4.38666575232725247094405384079, −4.03112471485534098047063447377, −2.97511834686345857111807767680, −2.34457380910952339347930140543, −1.14796370415559328739916504517,
1.14796370415559328739916504517, 2.34457380910952339347930140543, 2.97511834686345857111807767680, 4.03112471485534098047063447377, 4.38666575232725247094405384079, 5.22689540095913343017114111822, 5.78833706020384958605153062717, 6.49720552104566594219101665364, 7.44208403537851787648727669219, 7.84946360791647026742697521895
