L(s) = 1 | + 3.17·3-s − 4.23·5-s − 7-s + 7.08·9-s − 4.99·11-s − 0.439·13-s − 13.4·15-s + 3.50·17-s − 3.45·19-s − 3.17·21-s + 4.39·23-s + 12.9·25-s + 12.9·27-s + 0.132·29-s + 5.55·31-s − 15.8·33-s + 4.23·35-s + 0.572·37-s − 1.39·39-s + 5.42·41-s + 5.69·43-s − 30.0·45-s + 9.37·47-s + 49-s + 11.1·51-s + 3.65·53-s + 21.1·55-s + ⋯ |
L(s) = 1 | + 1.83·3-s − 1.89·5-s − 0.377·7-s + 2.36·9-s − 1.50·11-s − 0.121·13-s − 3.47·15-s + 0.849·17-s − 0.791·19-s − 0.693·21-s + 0.916·23-s + 2.59·25-s + 2.49·27-s + 0.0246·29-s + 0.998·31-s − 2.76·33-s + 0.716·35-s + 0.0941·37-s − 0.223·39-s + 0.847·41-s + 0.868·43-s − 4.47·45-s + 1.36·47-s + 0.142·49-s + 1.55·51-s + 0.502·53-s + 2.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.288648171\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.288648171\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 0.132T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 - 0.572T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 - 2.20T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 + 3.11T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.364044475666062321052054514763, −7.936615549433296388393738646327, −7.41782793489720060292790957328, −6.83354474451576507222623714695, −5.29523981380330527845545083208, −4.29590469504274238135791903395, −3.84042357554482567933012128604, −2.89841585752382476635408180733, −2.57591563725365522360178328446, −0.800771961243541812890219023487,
0.800771961243541812890219023487, 2.57591563725365522360178328446, 2.89841585752382476635408180733, 3.84042357554482567933012128604, 4.29590469504274238135791903395, 5.29523981380330527845545083208, 6.83354474451576507222623714695, 7.41782793489720060292790957328, 7.936615549433296388393738646327, 8.364044475666062321052054514763