L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 3·14-s − 15-s + 16-s + 17-s − 18-s − 20-s + 3·21-s − 22-s + 23-s − 24-s + 25-s + 27-s + 3·28-s − 9·29-s + 30-s + 5·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s + 0.654·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.566·28-s − 1.67·29-s + 0.182·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.985690828\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.985690828\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.200010519200582877899810655947, −7.55641133978648007448819218700, −7.11272009894724923730674981873, −6.09125932309452500302177068438, −5.24330738793665594056101069532, −4.35364167203949399678459138938, −3.65455134409835675353415636619, −2.61866266203842887435088817297, −1.79081673531795077309604892575, −0.847259894073213578695950316543,
0.847259894073213578695950316543, 1.79081673531795077309604892575, 2.61866266203842887435088817297, 3.65455134409835675353415636619, 4.35364167203949399678459138938, 5.24330738793665594056101069532, 6.09125932309452500302177068438, 7.11272009894724923730674981873, 7.55641133978648007448819218700, 8.200010519200582877899810655947