| L(s) = 1 | − 2·5-s + 2·13-s + 6·17-s + 19-s + 2·23-s − 25-s + 6·29-s − 31-s + 2·37-s − 2·41-s + 9·43-s + 2·47-s − 6·53-s + 8·59-s − 11·61-s − 4·65-s − 12·67-s + 4·71-s − 5·73-s − 4·79-s − 4·83-s − 12·85-s − 18·89-s − 2·95-s − 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s + 1.45·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s + 1.11·29-s − 0.179·31-s + 0.328·37-s − 0.312·41-s + 1.37·43-s + 0.291·47-s − 0.824·53-s + 1.04·59-s − 1.40·61-s − 0.496·65-s − 1.46·67-s + 0.474·71-s − 0.585·73-s − 0.450·79-s − 0.439·83-s − 1.30·85-s − 1.90·89-s − 0.205·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816023700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816023700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.50077734819321, −15.93842541946582, −15.49549182734934, −14.93995856407086, −14.15097508818575, −13.92497681128025, −12.97189869436174, −12.48530093390178, −11.89628007065678, −11.44158724772472, −10.76810643801839, −10.16359448478981, −9.531089993666978, −8.756383650574351, −8.213392570960040, −7.562571542436122, −7.167188903028490, −6.153374950149745, −5.681199351213655, −4.763800792449887, −4.130905224735126, −3.369906930449931, −2.814534529061591, −1.523576211196446, −0.6695664447886770,
0.6695664447886770, 1.523576211196446, 2.814534529061591, 3.369906930449931, 4.130905224735126, 4.763800792449887, 5.681199351213655, 6.153374950149745, 7.167188903028490, 7.562571542436122, 8.213392570960040, 8.756383650574351, 9.531089993666978, 10.16359448478981, 10.76810643801839, 11.44158724772472, 11.89628007065678, 12.48530093390178, 12.97189869436174, 13.92497681128025, 14.15097508818575, 14.93995856407086, 15.49549182734934, 15.93842541946582, 16.50077734819321
