L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 2.24·11-s − 12-s + 5.65·13-s − 15-s + 16-s + 2.58·17-s − 18-s − 6.82·19-s + 20-s − 2.24·22-s + 3.17·23-s + 24-s + 25-s − 5.65·26-s − 27-s + 2.58·29-s + 30-s − 10.2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.676·11-s − 0.288·12-s + 1.56·13-s − 0.258·15-s + 0.250·16-s + 0.627·17-s − 0.235·18-s − 1.56·19-s + 0.223·20-s − 0.478·22-s + 0.661·23-s + 0.204·24-s + 0.200·25-s − 1.10·26-s − 0.192·27-s + 0.480·29-s + 0.182·30-s − 1.83·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.198726479\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.198726479\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 0.343T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.07T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−9.280149518025642141406665033699, −8.959184756505256443497317599205, −7.977809534325356318101921351504, −7.02347788447525799477078133697, −6.18125565339828675029212663643, −5.77461662748256950288469817078, −4.42192514583588912725900418514, −3.43687336418807623598169474559, −1.97581342947159375154038880819, −0.942803363561806052611621428449,
0.942803363561806052611621428449, 1.97581342947159375154038880819, 3.43687336418807623598169474559, 4.42192514583588912725900418514, 5.77461662748256950288469817078, 6.18125565339828675029212663643, 7.02347788447525799477078133697, 7.977809534325356318101921351504, 8.959184756505256443497317599205, 9.280149518025642141406665033699