| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 6·11-s − 12-s − 13-s − 4·14-s + 16-s + 4·17-s − 18-s + 2·19-s − 4·21-s + 6·22-s + 6·23-s + 24-s + 26-s − 27-s + 4·28-s − 10·29-s + 4·31-s − 32-s + 6·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.872·21-s + 1.27·22-s + 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.104056630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104056630\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.254068378749180059371542633375, −8.136144422425248068944711105669, −7.74397499109575969153133695267, −7.20093415852113096973483335048, −5.80136496223537883339277826280, −5.28167187316235020398112518383, −4.56103262582491243050378997086, −3.04344583714730325251474537199, −1.99334892693085338905958438774, −0.820563498738137226109287832891,
0.820563498738137226109287832891, 1.99334892693085338905958438774, 3.04344583714730325251474537199, 4.56103262582491243050378997086, 5.28167187316235020398112518383, 5.80136496223537883339277826280, 7.20093415852113096973483335048, 7.74397499109575969153133695267, 8.136144422425248068944711105669, 9.254068378749180059371542633375
