L(s) = 1 | + 7-s + 6·11-s + 13-s + 3·17-s + 4·19-s + 3·23-s − 3·29-s − 5·31-s + 10·37-s − 9·41-s − 43-s + 49-s + 9·53-s + 9·59-s + 11·61-s − 4·67-s − 12·71-s + 10·73-s + 6·77-s + 10·79-s − 9·83-s + 6·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.625·23-s − 0.557·29-s − 0.898·31-s + 1.64·37-s − 1.40·41-s − 0.152·43-s + 1/7·49-s + 1.23·53-s + 1.17·59-s + 1.40·61-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.683·77-s + 1.12·79-s − 0.987·83-s + 0.635·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.405094989\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.405094989\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.16749242857127, −14.63435301563257, −14.55199977580649, −13.75405174238478, −13.29154834956472, −12.67011057164665, −11.90684998147792, −11.64150403583089, −11.21953349187978, −10.45610368516423, −9.752863841814162, −9.370786561835511, −8.774030728287253, −8.244993876985157, −7.446281039302665, −7.020733064821601, −6.390382701478395, −5.666919860268074, −5.195202222115868, −4.324678216224819, −3.733114121738830, −3.247466186051301, −2.199106002653414, −1.368088057924999, −0.8297656693209773,
0.8297656693209773, 1.368088057924999, 2.199106002653414, 3.247466186051301, 3.733114121738830, 4.324678216224819, 5.195202222115868, 5.666919860268074, 6.390382701478395, 7.020733064821601, 7.446281039302665, 8.244993876985157, 8.774030728287253, 9.370786561835511, 9.752863841814162, 10.45610368516423, 11.21953349187978, 11.64150403583089, 11.90684998147792, 12.67011057164665, 13.29154834956472, 13.75405174238478, 14.55199977580649, 14.63435301563257, 15.16749242857127