| L(s) = 1 | − 0.470·3-s + 5-s − 2.71·7-s − 2.77·9-s − 5.55·11-s − 2.02·13-s − 0.470·15-s − 3.77·17-s + 19-s + 1.28·21-s + 5.77·23-s + 25-s + 2.71·27-s + 5.66·29-s − 7.55·31-s + 2.61·33-s − 2.71·35-s + 3.75·37-s + 0.954·39-s − 12.6·41-s − 9.43·43-s − 2.77·45-s − 11.1·47-s + 0.397·49-s + 1.77·51-s + 8.85·53-s − 5.55·55-s + ⋯ |
| L(s) = 1 | − 0.271·3-s + 0.447·5-s − 1.02·7-s − 0.926·9-s − 1.67·11-s − 0.562·13-s − 0.121·15-s − 0.916·17-s + 0.229·19-s + 0.279·21-s + 1.20·23-s + 0.200·25-s + 0.523·27-s + 1.05·29-s − 1.35·31-s + 0.455·33-s − 0.459·35-s + 0.616·37-s + 0.152·39-s − 1.97·41-s − 1.43·43-s − 0.414·45-s − 1.62·47-s + 0.0567·49-s + 0.249·51-s + 1.21·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5955135815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5955135815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.470T + 3T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 5.66T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.242296849083155040410901995418, −7.09702755736915772647496201614, −6.76133110924091097130551935684, −5.84877752930302281723861764207, −5.22754637389266793592454876208, −4.75284882876662508869329710955, −3.29367484510882281022394477319, −2.88678675160056176430591975932, −2.03435629994206944625494656176, −0.37877980661816601640281592790,
0.37877980661816601640281592790, 2.03435629994206944625494656176, 2.88678675160056176430591975932, 3.29367484510882281022394477319, 4.75284882876662508869329710955, 5.22754637389266793592454876208, 5.84877752930302281723861764207, 6.76133110924091097130551935684, 7.09702755736915772647496201614, 8.242296849083155040410901995418
