| L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s + 0.438·7-s + 8-s − 9-s − 1.56i·11-s + i·12-s + (0.561 − 3.56i)13-s + 0.438·14-s + 16-s − 6.68i·17-s − 18-s − 7.68i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 0.165·7-s + 0.353·8-s − 0.333·9-s − 0.470i·11-s + 0.288i·12-s + (0.155 − 0.987i)13-s + 0.117·14-s + 0.250·16-s − 1.62i·17-s − 0.235·18-s − 1.76i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.629896188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.629896188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.561 + 3.56i)T \) |
| good | 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 1.56iT - 11T^{2} \) |
| 17 | \( 1 + 6.68iT - 17T^{2} \) |
| 19 | \( 1 + 7.68iT - 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 5.56iT - 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 4.56iT - 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 + 9.56iT - 59T^{2} \) |
| 61 | \( 1 + 0.684T + 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 5.12iT - 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.352600574802753441678392786747, −8.229869970507304107078254392717, −7.52559712146278870382681480927, −6.60808963845234723784652321178, −5.69032602622908653510423076280, −4.99598123622721561171684245662, −4.32978101891680160418753567586, −3.11793430939152688161468129420, −2.64100028273112984720574543324, −0.76595900528964349667836063529,
1.51095203082177497676708257270, 2.19384758200483174757776531505, 3.61211647204185011524979466040, 4.21435301830112040252756670200, 5.31611625451242825294091496093, 6.16301545003711982548648108591, 6.69814371711399276098846238320, 7.64009850115518200241426258725, 8.317967887543248637789949910668, 9.122077634205414766456704487144
