| L(s) = 1 | − 2·2-s + 4·4-s − 15.8·5-s − 8·8-s + 31.6·10-s − 51.8·11-s − 38.8·13-s + 16·16-s + 27.3·17-s − 76.5·19-s − 63.3·20-s + 103.·22-s − 147.·23-s + 125.·25-s + 77.6·26-s − 240.·29-s + 296.·31-s − 32·32-s − 54.6·34-s − 161.·37-s + 153.·38-s + 126.·40-s − 102.·41-s − 328.·43-s − 207.·44-s + 294.·46-s + 67.9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.41·5-s − 0.353·8-s + 1.00·10-s − 1.42·11-s − 0.828·13-s + 0.250·16-s + 0.390·17-s − 0.923·19-s − 0.708·20-s + 1.00·22-s − 1.33·23-s + 1.00·25-s + 0.585·26-s − 1.53·29-s + 1.71·31-s − 0.176·32-s − 0.275·34-s − 0.717·37-s + 0.653·38-s + 0.500·40-s − 0.392·41-s − 1.16·43-s − 0.710·44-s + 0.944·46-s + 0.210·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2385219501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2385219501\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 15.8T + 125T^{2} \) |
| 11 | \( 1 + 51.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 461.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 130.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 181.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 409.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 347.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
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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
−9.920760678026263954522944325170, −8.687005971344750862646124584750, −7.893452890775865171720399343145, −7.65111755265060495970696936680, −6.55687947172848721711430682831, −5.32852797218403187572843427200, −4.30722919275791766187054369466, −3.21461499887703329325054823246, −2.09407121541824752779551192959, −0.27849165999967960918650292112,
0.27849165999967960918650292112, 2.09407121541824752779551192959, 3.21461499887703329325054823246, 4.30722919275791766187054369466, 5.32852797218403187572843427200, 6.55687947172848721711430682831, 7.65111755265060495970696936680, 7.893452890775865171720399343145, 8.687005971344750862646124584750, 9.920760678026263954522944325170
