L(s) = 1 | − 1.34·3-s − 0.833·7-s − 1.18·9-s − 0.833·11-s − 4.56·13-s − 5.45·17-s + 8.65·19-s + 1.12·21-s + 3.53·23-s + 5.63·27-s + 3.26·29-s − 6.00·31-s + 1.12·33-s − 7.31·37-s + 6.15·39-s + 1.86·41-s − 1.63·43-s − 0.833·47-s − 6.30·49-s + 7.35·51-s − 6.42·53-s − 11.6·57-s − 13.5·59-s + 5.88·61-s + 0.984·63-s + 1.49·67-s − 4.76·69-s + ⋯ |
L(s) = 1 | − 0.778·3-s − 0.314·7-s − 0.393·9-s − 0.251·11-s − 1.26·13-s − 1.32·17-s + 1.98·19-s + 0.245·21-s + 0.736·23-s + 1.08·27-s + 0.607·29-s − 1.07·31-s + 0.195·33-s − 1.20·37-s + 0.985·39-s + 0.291·41-s − 0.249·43-s − 0.121·47-s − 0.900·49-s + 1.02·51-s − 0.882·53-s − 1.54·57-s − 1.76·59-s + 0.753·61-s + 0.124·63-s + 0.183·67-s − 0.573·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6271241866\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6271241866\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 + 0.833T + 7T^{2} \) |
| 11 | \( 1 + 0.833T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 - 8.65T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 0.833T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 5.88T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.45321260039410978104404576580, −6.94594081939532416366924066712, −6.35413232769941600572470876888, −5.37135069101951160079606405748, −5.16242252188063419691036240488, −4.39917272164869624707321581506, −3.21625065019350257804145380109, −2.75865822108879107173767190997, −1.62567419204546333727487084347, −0.38301194296712549894306772612,
0.38301194296712549894306772612, 1.62567419204546333727487084347, 2.75865822108879107173767190997, 3.21625065019350257804145380109, 4.39917272164869624707321581506, 5.16242252188063419691036240488, 5.37135069101951160079606405748, 6.35413232769941600572470876888, 6.94594081939532416366924066712, 7.45321260039410978104404576580