| L(s) = 1 | − 1.84·2-s + 0.727·3-s − 4.59·4-s + 12.1·5-s − 1.34·6-s − 17.8·7-s + 23.2·8-s − 26.4·9-s − 22.4·10-s + 16.9·11-s − 3.34·12-s − 51.5·13-s + 32.9·14-s + 8.84·15-s − 6.07·16-s + 120.·17-s + 48.8·18-s − 5.77·19-s − 55.9·20-s − 12.9·21-s − 31.3·22-s + 210.·23-s + 16.9·24-s + 22.8·25-s + 95.0·26-s − 38.9·27-s + 82.0·28-s + ⋯ |
| L(s) = 1 | − 0.652·2-s + 0.140·3-s − 0.574·4-s + 1.08·5-s − 0.0913·6-s − 0.963·7-s + 1.02·8-s − 0.980·9-s − 0.709·10-s + 0.465·11-s − 0.0804·12-s − 1.09·13-s + 0.628·14-s + 0.152·15-s − 0.0949·16-s + 1.72·17-s + 0.639·18-s − 0.0697·19-s − 0.624·20-s − 0.134·21-s − 0.303·22-s + 1.90·23-s + 0.143·24-s + 0.182·25-s + 0.716·26-s − 0.277·27-s + 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.100085813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100085813\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.84T + 8T^{2} \) |
| 3 | \( 1 - 0.727T + 27T^{2} \) |
| 5 | \( 1 - 12.1T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 11 | \( 1 - 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.77T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 233.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 92.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 190.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 62.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 564.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 339.T + 9.12e5T^{2} \) |
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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
−9.132133605897125245947396082056, −8.337206193549453647581276965717, −7.34740086624543874184670802351, −6.63572505034376283259720957033, −5.38593575948155311569335746910, −5.27470370807550461903520678113, −3.66140936032268083594898707023, −2.89945042071412760267683175387, −1.69587992194554997920748684777, −0.54551140686739750006272621768,
0.54551140686739750006272621768, 1.69587992194554997920748684777, 2.89945042071412760267683175387, 3.66140936032268083594898707023, 5.27470370807550461903520678113, 5.38593575948155311569335746910, 6.63572505034376283259720957033, 7.34740086624543874184670802351, 8.337206193549453647581276965717, 9.132133605897125245947396082056
