L(s) = 1 | − 3-s + 7-s + 9-s − 11-s − 6·13-s − 7·19-s − 21-s − 23-s − 5·25-s − 27-s − 10·29-s + 10·31-s + 33-s + 6·37-s + 6·39-s − 5·41-s − 4·43-s + 47-s + 49-s − 3·53-s + 7·57-s + 3·59-s + 9·61-s + 63-s + 4·67-s + 69-s + 12·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 1.60·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.192·27-s − 1.85·29-s + 1.79·31-s + 0.174·33-s + 0.986·37-s + 0.960·39-s − 0.780·41-s − 0.609·43-s + 0.145·47-s + 1/7·49-s − 0.412·53-s + 0.927·57-s + 0.390·59-s + 1.15·61-s + 0.125·63-s + 0.488·67-s + 0.120·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8852282548\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8852282548\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.86689573739229326670331759203, −7.15738836480976018426634785697, −6.45823797789105031950808801625, −5.76724216636908825407876159501, −4.98940880259313026642751324635, −4.49500928870921479999764809923, −3.67811872117377206431297311557, −2.42044931083738728688813152719, −1.94253406055734435080975540115, −0.45928729866942828116686353310,
0.45928729866942828116686353310, 1.94253406055734435080975540115, 2.42044931083738728688813152719, 3.67811872117377206431297311557, 4.49500928870921479999764809923, 4.98940880259313026642751324635, 5.76724216636908825407876159501, 6.45823797789105031950808801625, 7.15738836480976018426634785697, 7.86689573739229326670331759203