L(s) = 1 | + 1.61·3-s − 0.618·7-s − 0.381·9-s + 3.23·11-s + 3.23·13-s + 2.76·17-s − 1.23·19-s − 1.00·21-s + 3.38·23-s − 5.47·27-s + 2.85·29-s + 7.23·31-s + 5.23·33-s + 6·37-s + 5.23·39-s − 8.85·41-s + 11.3·43-s − 2.09·47-s − 6.61·49-s + 4.47·51-s + 12.9·53-s − 2.00·57-s − 8.18·59-s − 13.0·61-s + 0.236·63-s + 1.52·67-s + 5.47·69-s + ⋯ |
L(s) = 1 | + 0.934·3-s − 0.233·7-s − 0.127·9-s + 0.975·11-s + 0.897·13-s + 0.670·17-s − 0.283·19-s − 0.218·21-s + 0.705·23-s − 1.05·27-s + 0.529·29-s + 1.29·31-s + 0.911·33-s + 0.986·37-s + 0.838·39-s − 1.38·41-s + 1.72·43-s − 0.304·47-s − 0.945·49-s + 0.626·51-s + 1.77·53-s − 0.264·57-s − 1.06·59-s − 1.67·61-s + 0.0297·63-s + 0.186·67-s + 0.658·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.289681487\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.289681487\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 8.85T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 8.18T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.773478730230016097609468316969, −9.025184571087050818542133128137, −8.461989664957623573383504218483, −7.63957013030459316830522921997, −6.56176963931508687891396831498, −5.84429206238643508853307198227, −4.47855642636661809533283213822, −3.52228700016307486411394363827, −2.72378513451019937579751996738, −1.26646406580910457750403388417,
1.26646406580910457750403388417, 2.72378513451019937579751996738, 3.52228700016307486411394363827, 4.47855642636661809533283213822, 5.84429206238643508853307198227, 6.56176963931508687891396831498, 7.63957013030459316830522921997, 8.461989664957623573383504218483, 9.025184571087050818542133128137, 9.773478730230016097609468316969