| L(s) = 1 | + 2.82·7-s − 4.89·11-s − 4.89·13-s + 3.46·17-s + 6.92·19-s + 4·23-s + 8.48·29-s − 6.92·31-s − 4.89·37-s + 5.65·41-s − 11.3·43-s + 4·47-s + 1.00·49-s + 3.46·53-s − 4.89·59-s − 6·61-s + 5.65·67-s + 9.79·71-s − 13.8·77-s + 6.92·79-s + 16·83-s − 13.8·91-s + 9.79·97-s − 8.48·101-s + 2.82·103-s + 8·107-s − 6·109-s + ⋯ |
| L(s) = 1 | + 1.06·7-s − 1.47·11-s − 1.35·13-s + 0.840·17-s + 1.58·19-s + 0.834·23-s + 1.57·29-s − 1.24·31-s − 0.805·37-s + 0.883·41-s − 1.72·43-s + 0.583·47-s + 0.142·49-s + 0.475·53-s − 0.637·59-s − 0.768·61-s + 0.691·67-s + 1.16·71-s − 1.57·77-s + 0.779·79-s + 1.75·83-s − 1.45·91-s + 0.994·97-s − 0.844·101-s + 0.278·103-s + 0.773·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.018768128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018768128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−7.70118836692553693933360464926, −7.54944964169661102653198705336, −6.68338139365426628721701359865, −5.31992298486176585851727282478, −5.27731381307781240473338933276, −4.65758268255492623478247246416, −3.37295562842767001988830913227, −2.74065624879688752681304813168, −1.83846525590925940165874525461, −0.72095540930264825897670539789,
0.72095540930264825897670539789, 1.83846525590925940165874525461, 2.74065624879688752681304813168, 3.37295562842767001988830913227, 4.65758268255492623478247246416, 5.27731381307781240473338933276, 5.31992298486176585851727282478, 6.68338139365426628721701359865, 7.54944964169661102653198705336, 7.70118836692553693933360464926
