| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 6.24·11-s − 12-s − 5.65·13-s − 15-s + 16-s + 5.41·17-s − 18-s − 1.17·19-s + 20-s + 6.24·22-s + 8.82·23-s + 24-s + 25-s + 5.65·26-s − 27-s + 5.41·29-s + 30-s − 1.75·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.88·11-s − 0.288·12-s − 1.56·13-s − 0.258·15-s + 0.250·16-s + 1.31·17-s − 0.235·18-s − 0.268·19-s + 0.223·20-s + 1.33·22-s + 1.84·23-s + 0.204·24-s + 0.200·25-s + 1.10·26-s − 0.192·27-s + 1.00·29-s + 0.182·30-s − 0.315·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8354616403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8354616403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.733285241697315604767741879971, −8.759756208564014220552681014969, −7.70435008066193203384493680290, −7.35019609921314411283933631029, −6.29792174281127491510778634348, −5.21833765073136983848455326884, −4.95211788084688748619227352142, −3.06859717288621484091304238475, −2.29712506192564374505929072677, −0.72516733990553923785407835706,
0.72516733990553923785407835706, 2.29712506192564374505929072677, 3.06859717288621484091304238475, 4.95211788084688748619227352142, 5.21833765073136983848455326884, 6.29792174281127491510778634348, 7.35019609921314411283933631029, 7.70435008066193203384493680290, 8.759756208564014220552681014969, 9.733285241697315604767741879971
