| L(s) = 1 | + 1.56·2-s − 5.56·4-s + 5·5-s − 21.1·8-s + 7.80·10-s + 10.2·11-s − 34.3·13-s + 11.4·16-s − 82.6·17-s − 90.7·19-s − 27.8·20-s + 16·22-s + 12.1·23-s + 25·25-s − 53.6·26-s + 105.·29-s + 142.·31-s + 187.·32-s − 128.·34-s + 64.8·37-s − 141.·38-s − 105.·40-s − 195.·41-s − 319.·43-s − 56.9·44-s + 18.9·46-s − 318.·47-s + ⋯ |
| L(s) = 1 | + 0.552·2-s − 0.695·4-s + 0.447·5-s − 0.935·8-s + 0.246·10-s + 0.280·11-s − 0.732·13-s + 0.178·16-s − 1.17·17-s − 1.09·19-s − 0.310·20-s + 0.155·22-s + 0.110·23-s + 0.200·25-s − 0.404·26-s + 0.673·29-s + 0.823·31-s + 1.03·32-s − 0.650·34-s + 0.288·37-s − 0.604·38-s − 0.418·40-s − 0.743·41-s − 1.13·43-s − 0.195·44-s + 0.0607·46-s − 0.989·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.704147006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704147006\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 11 | \( 1 - 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 12.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 64.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 284T + 2.05e5T^{2} \) |
| 61 | \( 1 - 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 549.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 556.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 376.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 752.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 945.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 180.T + 9.12e5T^{2} \) |
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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
−8.615669214236221593066922022917, −8.209871480198967937135438088196, −6.77783948051700661574075855378, −6.42446368842428570407377376871, −5.32566981380616519607906216617, −4.70077979524950616449964202921, −4.00410992233582864127978480211, −2.90997972760684705013016309415, −1.98413076513998612259315007903, −0.52704134419363556317130216397,
0.52704134419363556317130216397, 1.98413076513998612259315007903, 2.90997972760684705013016309415, 4.00410992233582864127978480211, 4.70077979524950616449964202921, 5.32566981380616519607906216617, 6.42446368842428570407377376871, 6.77783948051700661574075855378, 8.209871480198967937135438088196, 8.615669214236221593066922022917
