L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s + 4·22-s − 8·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.78270031849701, −12.55994953230907, −12.06659666146859, −11.60777870602924, −11.16287848516810, −10.32766211100919, −10.08275536098445, −9.639097520462284, −9.257772354570830, −8.935935696661124, −8.128948835137126, −7.604707178607630, −7.292678014077952, −6.684845677154928, −6.109790544640027, −5.931819908988691, −5.514732374953802, −4.663908347091673, −4.074635728361258, −3.652952578597312, −3.283487545400247, −2.861872048761030, −2.154053341321339, −1.488856806822536, −1.024210034184673, 0,
1.024210034184673, 1.488856806822536, 2.154053341321339, 2.861872048761030, 3.283487545400247, 3.652952578597312, 4.074635728361258, 4.663908347091673, 5.514732374953802, 5.931819908988691, 6.109790544640027, 6.684845677154928, 7.292678014077952, 7.604707178607630, 8.128948835137126, 8.935935696661124, 9.257772354570830, 9.639097520462284, 10.08275536098445, 10.32766211100919, 11.16287848516810, 11.60777870602924, 12.06659666146859, 12.55994953230907, 12.78270031849701