L(s) = 1 | − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s − 1.73·11-s − 1.73·12-s − 13-s + 16-s − 17-s − 1.99·18-s + 1.73·22-s + 1.73·24-s + 25-s + 26-s − 1.73·27-s − 32-s + 2.99·33-s + 34-s + 1.99·36-s + 1.73·39-s − 1.73·44-s − 1.73·48-s − 50-s + 1.73·51-s − 52-s + ⋯ |
L(s) = 1 | − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s − 1.73·11-s − 1.73·12-s − 13-s + 16-s − 17-s − 1.99·18-s + 1.73·22-s + 1.73·24-s + 25-s + 26-s − 1.73·27-s − 32-s + 2.99·33-s + 34-s + 1.99·36-s + 1.73·39-s − 1.73·44-s − 1.73·48-s − 50-s + 1.73·51-s − 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2422213340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2422213340\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.836511413520278058370010727934, −7.933317147152552473405151975889, −7.23348364605972382626051290849, −6.69149825291659222536667894203, −5.86221188537596506521788798599, −5.16527165361072307011411089017, −4.56821751285816804460011112535, −2.94083115754563720068588777951, −1.98330077646091093478117271983, −0.51974453057723587216119713507,
0.51974453057723587216119713507, 1.98330077646091093478117271983, 2.94083115754563720068588777951, 4.56821751285816804460011112535, 5.16527165361072307011411089017, 5.86221188537596506521788798599, 6.69149825291659222536667894203, 7.23348364605972382626051290849, 7.933317147152552473405151975889, 8.836511413520278058370010727934