L(s) = 1 | + 5-s + 7-s − 5·11-s + 5·13-s + 7·17-s + 2·19-s + 2·23-s + 25-s + 7·29-s + 4·31-s + 35-s + 6·37-s + 12·41-s + 2·43-s − 47-s + 49-s − 5·55-s − 4·59-s − 4·61-s + 5·65-s − 8·67-s + 6·73-s − 5·77-s − 3·79-s − 4·83-s + 7·85-s + 5·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.50·11-s + 1.38·13-s + 1.69·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.29·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 1.87·41-s + 0.304·43-s − 0.145·47-s + 1/7·49-s − 0.674·55-s − 0.520·59-s − 0.512·61-s + 0.620·65-s − 0.977·67-s + 0.702·73-s − 0.569·77-s − 0.337·79-s − 0.439·83-s + 0.759·85-s + 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.237848191\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.237848191\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.73593808060138, −15.17341741898694, −14.44846201485236, −13.95902906747386, −13.58524261386504, −12.86268932318989, −12.54889780585635, −11.75478260937970, −11.19313560428539, −10.56120496398044, −10.24113554599349, −9.581789404081353, −8.900723508103308, −8.218012091983444, −7.790536168299867, −7.322548635576148, −6.153587864700317, −5.999524131316417, −5.204734862068180, −4.707383041058693, −3.805852106112432, −2.952171059503247, −2.576696915809177, −1.349291651369208, −0.8433102132721604,
0.8433102132721604, 1.349291651369208, 2.576696915809177, 2.952171059503247, 3.805852106112432, 4.707383041058693, 5.204734862068180, 5.999524131316417, 6.153587864700317, 7.322548635576148, 7.790536168299867, 8.218012091983444, 8.900723508103308, 9.581789404081353, 10.24113554599349, 10.56120496398044, 11.19313560428539, 11.75478260937970, 12.54889780585635, 12.86268932318989, 13.58524261386504, 13.95902906747386, 14.44846201485236, 15.17341741898694, 15.73593808060138