| L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s + 2·11-s − 2·12-s + 5·13-s + 4·16-s − 17-s + 2·19-s − 21-s + 23-s + 27-s + 2·28-s + 8·29-s + 31-s + 2·33-s − 2·36-s + 3·37-s + 5·39-s − 7·41-s − 4·44-s + 47-s + 4·48-s + 49-s − 51-s − 10·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.38·13-s + 16-s − 0.242·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.377·28-s + 1.48·29-s + 0.179·31-s + 0.348·33-s − 1/3·36-s + 0.493·37-s + 0.800·39-s − 1.09·41-s − 0.603·44-s + 0.145·47-s + 0.577·48-s + 1/7·49-s − 0.140·51-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253050443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253050443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.960914551292045289568078565561, −7.08068842893042573157127957586, −6.34121153574682603383165408953, −5.74686663003999979672783894966, −4.76901210207587113670938973958, −4.18929266641064432537778959756, −3.47631589231825098224073927385, −2.90670913883543145720496370922, −1.56510596663840806655535665544, −0.77036299586008536961730007399,
0.77036299586008536961730007399, 1.56510596663840806655535665544, 2.90670913883543145720496370922, 3.47631589231825098224073927385, 4.18929266641064432537778959756, 4.76901210207587113670938973958, 5.74686663003999979672783894966, 6.34121153574682603383165408953, 7.08068842893042573157127957586, 7.960914551292045289568078565561
