| L(s) = 1 | − 2.23·2-s + 1.90·3-s + 3.00·4-s + 4.23·5-s − 4.26·6-s + 7-s − 2.25·8-s + 0.626·9-s − 9.47·10-s − 0.780·11-s + 5.72·12-s − 5.45·13-s − 2.23·14-s + 8.06·15-s − 0.966·16-s + 1.79·17-s − 1.40·18-s − 4.44·19-s + 12.7·20-s + 1.90·21-s + 1.74·22-s + 7.54·23-s − 4.29·24-s + 12.9·25-s + 12.1·26-s − 4.52·27-s + 3.00·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.09·3-s + 1.50·4-s + 1.89·5-s − 1.73·6-s + 0.377·7-s − 0.797·8-s + 0.208·9-s − 2.99·10-s − 0.235·11-s + 1.65·12-s − 1.51·13-s − 0.598·14-s + 2.08·15-s − 0.241·16-s + 0.436·17-s − 0.330·18-s − 1.02·19-s + 2.84·20-s + 0.415·21-s + 0.372·22-s + 1.57·23-s − 0.877·24-s + 2.58·25-s + 2.39·26-s − 0.869·27-s + 0.568·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.016795120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016795120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 + 0.780T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 + 2.02T + 37T^{2} \) |
| 41 | \( 1 - 9.44T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 4.69T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 + 6.49T + 79T^{2} \) |
| 83 | \( 1 + 0.00279T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−8.312329474999551595673487759714, −7.65801995670187326443065765508, −6.86227804251204009352617244483, −6.30130706124821000490178811310, −5.23808896464111024795911457819, −4.62410316426306931379761494655, −2.86642903015480479211223456228, −2.52299764248025546819610844780, −1.88251986395840615263202689883, −0.914118775405580418287630417406,
0.914118775405580418287630417406, 1.88251986395840615263202689883, 2.52299764248025546819610844780, 2.86642903015480479211223456228, 4.62410316426306931379761494655, 5.23808896464111024795911457819, 6.30130706124821000490178811310, 6.86227804251204009352617244483, 7.65801995670187326443065765508, 8.312329474999551595673487759714
