L(s) = 1 | − 0.687·3-s + 1.01·7-s − 2.52·9-s − 5.12·11-s − 6.08·13-s − 3.19·17-s − 3.42·19-s − 0.695·21-s + 2.91·23-s + 3.79·27-s − 1.55·29-s + 7.99·31-s + 3.52·33-s − 8.40·37-s + 4.18·39-s − 1.86·41-s − 5.22·43-s − 4.80·47-s − 5.97·49-s + 2.19·51-s − 10.0·53-s + 2.35·57-s − 2.89·59-s − 2.30·61-s − 2.55·63-s + 4.64·67-s − 2.00·69-s + ⋯ |
L(s) = 1 | − 0.396·3-s + 0.382·7-s − 0.842·9-s − 1.54·11-s − 1.68·13-s − 0.774·17-s − 0.786·19-s − 0.151·21-s + 0.608·23-s + 0.731·27-s − 0.288·29-s + 1.43·31-s + 0.612·33-s − 1.38·37-s + 0.669·39-s − 0.291·41-s − 0.796·43-s − 0.700·47-s − 0.853·49-s + 0.307·51-s − 1.38·53-s + 0.312·57-s − 0.376·59-s − 0.295·61-s − 0.322·63-s + 0.567·67-s − 0.241·69-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
\(L(1)\) |
\(\approx\) |
\(0.3429134907\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3429134907\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.687T + 3T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 6.08T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 7.99T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 0.595T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.81439485048123026349970261828, −6.85787130726983541481908270007, −6.40781780875906922540939916219, −5.33773735219881901383262385560, −5.02056393450835265312211941818, −4.52101384971990475871721572260, −3.17317151694594474305260049496, −2.60839895518116391784141701561, −1.87467895337901913100457728307, −0.26151085613594820245784517116,
0.26151085613594820245784517116, 1.87467895337901913100457728307, 2.60839895518116391784141701561, 3.17317151694594474305260049496, 4.52101384971990475871721572260, 5.02056393450835265312211941818, 5.33773735219881901383262385560, 6.40781780875906922540939916219, 6.85787130726983541481908270007, 7.81439485048123026349970261828