| L(s) = 1 | + 2.11e7·5-s + 7.32e8·7-s + 5.59e9·11-s − 6.30e10·13-s − 1.35e13·17-s + 1.39e13·19-s + 2.80e14·23-s − 2.92e13·25-s − 1.19e15·29-s + 3.88e15·31-s + 1.55e16·35-s + 2.72e16·37-s + 6.89e16·41-s + 3.24e16·43-s + 2.07e17·47-s − 2.12e16·49-s + 6.38e17·53-s + 1.18e17·55-s + 3.04e18·59-s − 5.64e18·61-s − 1.33e18·65-s − 3.96e18·67-s + 2.61e19·71-s + 1.37e19·73-s + 4.10e18·77-s − 1.18e20·79-s + 1.60e19·83-s + ⋯ |
| L(s) = 1 | + 0.968·5-s + 0.980·7-s + 0.0650·11-s − 0.126·13-s − 1.63·17-s + 0.520·19-s + 1.41·23-s − 0.0614·25-s − 0.528·29-s + 0.851·31-s + 0.950·35-s + 0.929·37-s + 0.802·41-s + 0.229·43-s + 0.575·47-s − 0.0381·49-s + 0.501·53-s + 0.0630·55-s + 0.775·59-s − 1.01·61-s − 0.122·65-s − 0.265·67-s + 0.954·71-s + 0.374·73-s + 0.0638·77-s − 1.40·79-s + 0.113·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.421190713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421190713\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.11e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 7.32e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 5.59e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.30e10T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.35e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.39e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.80e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.19e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.88e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.72e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 6.89e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 3.24e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.07e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 6.38e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.04e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.64e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 3.96e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.61e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.37e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.18e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.60e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.30e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.72e20T + 5.27e41T^{2} \) |
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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
−10.79630178307804610561012616688, −9.555875921991535762009789280736, −8.712585718763667026146533992921, −7.45594805869319674188445065429, −6.31384603160852270927695272036, −5.22317553352015517732649851995, −4.32087559981441605241883336242, −2.68353203388035437608203798335, −1.82998475824443860127466662482, −0.794448246286980641991351496254,
0.794448246286980641991351496254, 1.82998475824443860127466662482, 2.68353203388035437608203798335, 4.32087559981441605241883336242, 5.22317553352015517732649851995, 6.31384603160852270927695272036, 7.45594805869319674188445065429, 8.712585718763667026146533992921, 9.555875921991535762009789280736, 10.79630178307804610561012616688
