| L(s) = 1 | − 1.56·3-s + 5-s + 3.50·7-s − 0.561·9-s + 0.659·11-s + 5.91·13-s − 1.56·15-s + 0.844·17-s − 0.659·19-s − 5.47·21-s + 23-s + 25-s + 5.56·27-s + 1.59·29-s − 6.75·31-s − 1.02·33-s + 3.50·35-s − 11.7·37-s − 9.22·39-s + 6.40·41-s + 9.47·43-s − 0.561·45-s − 6.88·47-s + 5.27·49-s − 1.31·51-s + 6.64·53-s + 0.659·55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s + 1.32·7-s − 0.187·9-s + 0.198·11-s + 1.63·13-s − 0.403·15-s + 0.204·17-s − 0.151·19-s − 1.19·21-s + 0.208·23-s + 0.200·25-s + 1.07·27-s + 0.295·29-s − 1.21·31-s − 0.179·33-s + 0.592·35-s − 1.93·37-s − 1.47·39-s + 1.00·41-s + 1.44·43-s − 0.0837·45-s − 1.00·47-s + 0.754·49-s − 0.184·51-s + 0.913·53-s + 0.0889·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.698500895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698500895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 0.659T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 0.844T + 17T^{2} \) |
| 19 | \( 1 + 0.659T + 19T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 - 5.78T + 61T^{2} \) |
| 67 | \( 1 + 8.31T + 67T^{2} \) |
| 71 | \( 1 - 3.63T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
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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
−8.982675253708169267097441012434, −8.639951976706135274805001191644, −7.68279564422352762120401154319, −6.71069265155203779856264617423, −5.85677648523901971095267999764, −5.39380599875681564164645735690, −4.49242654461457814546483828129, −3.44344057798255965365626998244, −1.96785464371632331764879492860, −0.996302207714653502888639586750,
0.996302207714653502888639586750, 1.96785464371632331764879492860, 3.44344057798255965365626998244, 4.49242654461457814546483828129, 5.39380599875681564164645735690, 5.85677648523901971095267999764, 6.71069265155203779856264617423, 7.68279564422352762120401154319, 8.639951976706135274805001191644, 8.982675253708169267097441012434
