| L(s) = 1 | − 8.50·3-s + 5·5-s − 14.9·7-s + 45.3·9-s + 45.9·11-s + 76.1·13-s − 42.5·15-s + 105.·17-s + 93.8·19-s + 127.·21-s + 23·23-s + 25·25-s − 156.·27-s + 218.·29-s + 178.·31-s − 391.·33-s − 74.8·35-s − 85.2·37-s − 648.·39-s + 452.·41-s + 60.6·43-s + 226.·45-s − 232.·47-s − 118.·49-s − 895.·51-s − 352.·53-s + 229.·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.447·5-s − 0.808·7-s + 1.68·9-s + 1.26·11-s + 1.62·13-s − 0.732·15-s + 1.50·17-s + 1.13·19-s + 1.32·21-s + 0.208·23-s + 0.200·25-s − 1.11·27-s + 1.39·29-s + 1.03·31-s − 2.06·33-s − 0.361·35-s − 0.378·37-s − 2.66·39-s + 1.72·41-s + 0.215·43-s + 0.751·45-s − 0.723·47-s − 0.346·49-s − 2.45·51-s − 0.914·53-s + 0.563·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.892536441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892536441\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 8.50T + 27T^{2} \) |
| 7 | \( 1 + 14.9T + 343T^{2} \) |
| 11 | \( 1 - 45.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 452.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 232.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 352.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 445.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 171.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 297.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 494.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 220.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.34e3T + 9.12e5T^{2} \) |
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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
−9.128727558672231962297396794484, −8.044376661138430946108730435362, −6.86504423093014244255680554427, −6.33614329558116492172290881599, −5.87512197654250092590433400135, −5.06701857890292479740654926536, −3.97698238787436998022162073138, −3.12757028251538714564186688032, −1.18783338827593661531499877375, −0.916892722105821560862071458949,
0.916892722105821560862071458949, 1.18783338827593661531499877375, 3.12757028251538714564186688032, 3.97698238787436998022162073138, 5.06701857890292479740654926536, 5.87512197654250092590433400135, 6.33614329558116492172290881599, 6.86504423093014244255680554427, 8.044376661138430946108730435362, 9.128727558672231962297396794484
