| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 12-s + 2·15-s + 16-s + 18-s − 8·19-s + 2·20-s + 24-s + 3·25-s + 27-s + 12·29-s + 2·30-s + 32-s + 36-s − 8·38-s + 2·40-s − 8·43-s + 2·45-s + 48-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.204·24-s + 3/5·25-s + 0.192·27-s + 2.22·29-s + 0.365·30-s + 0.176·32-s + 1/6·36-s − 1.29·38-s + 0.316·40-s − 1.21·43-s + 0.298·45-s + 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.286902394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.286902394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.880541566397139490214892173475, −9.082506062594783472831133921500, −8.644229019078172602684577487278, −8.307525687782695431948900879973, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −5.48025531888955534300415720953, −4.84143328212506167706839183620, −4.32687879913691587957737883725, −3.69936628544405197022333195268, −2.74628747875934588873509462774, −2.39041205999551187286748903960, −1.43294542786364492056174526795,
1.43294542786364492056174526795, 2.39041205999551187286748903960, 2.74628747875934588873509462774, 3.69936628544405197022333195268, 4.32687879913691587957737883725, 4.84143328212506167706839183620, 5.48025531888955534300415720953, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 8.307525687782695431948900879973, 8.644229019078172602684577487278, 9.082506062594783472831133921500, 9.880541566397139490214892173475
