| L(s) = 1 | + 2.24·2-s + 3.04·4-s + 1.44·5-s + 2.04·7-s + 2.35·8-s + 3.24·10-s + 2.55·11-s + 4.60·14-s − 0.801·16-s + 5.29·17-s − 5.85·19-s + 4.40·20-s + 5.74·22-s + 1.89·23-s − 2.91·25-s + 6.24·28-s − 2.26·29-s − 4.26·31-s − 6.51·32-s + 11.8·34-s + 2.96·35-s + 5.35·37-s − 13.1·38-s + 3.40·40-s − 1.27·41-s + 6.13·43-s + 7.78·44-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.52·4-s + 0.646·5-s + 0.774·7-s + 0.833·8-s + 1.02·10-s + 0.770·11-s + 1.23·14-s − 0.200·16-s + 1.28·17-s − 1.34·19-s + 0.985·20-s + 1.22·22-s + 0.394·23-s − 0.582·25-s + 1.18·28-s − 0.421·29-s − 0.766·31-s − 1.15·32-s + 2.04·34-s + 0.500·35-s + 0.880·37-s − 2.13·38-s + 0.538·40-s − 0.198·41-s + 0.935·43-s + 1.17·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.057630296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.057630296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 - 6.13T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 0.576T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−9.528960750273212832454954758078, −8.646835769291802007558562425053, −7.62240315678215930170706087622, −6.71323377198385778553817496483, −5.88204919487585002572159186956, −5.38071371527214669348189380118, −4.38090465887099137820399437027, −3.72992797965384587137429162895, −2.53834240042012567804284135719, −1.56760851384533193583793769691,
1.56760851384533193583793769691, 2.53834240042012567804284135719, 3.72992797965384587137429162895, 4.38090465887099137820399437027, 5.38071371527214669348189380118, 5.88204919487585002572159186956, 6.71323377198385778553817496483, 7.62240315678215930170706087622, 8.646835769291802007558562425053, 9.528960750273212832454954758078
