L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.23·7-s − 8-s + 9-s + 5.30·11-s − 12-s − 5.90·13-s + 3.23·14-s + 16-s − 1.02·17-s − 18-s − 4.50·19-s + 3.23·21-s − 5.30·22-s − 1.92·23-s + 24-s + 5.90·26-s − 27-s − 3.23·28-s − 0.260·29-s − 2.24·31-s − 32-s − 5.30·33-s + 1.02·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s + 1.59·11-s − 0.288·12-s − 1.63·13-s + 0.863·14-s + 0.250·16-s − 0.248·17-s − 0.235·18-s − 1.03·19-s + 0.705·21-s − 1.13·22-s − 0.402·23-s + 0.204·24-s + 1.15·26-s − 0.192·27-s − 0.610·28-s − 0.0484·29-s − 0.402·31-s − 0.176·32-s − 0.922·33-s + 0.175·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5878580548\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5878580548\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 0.260T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 - 5.30T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 2.31T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 + 9.29T + 67T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + 0.733T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.803923550353660546531587111530, −7.57554585883996408500878316989, −7.04235951318806922424719211684, −6.34454139120765680077275474847, −5.90879720848716243908575444234, −4.63910180122538718613917457184, −3.92218329568021570395107037013, −2.83171450164717666645008857327, −1.84761573707582582185933118066, −0.49434924355337649794413937413,
0.49434924355337649794413937413, 1.84761573707582582185933118066, 2.83171450164717666645008857327, 3.92218329568021570395107037013, 4.63910180122538718613917457184, 5.90879720848716243908575444234, 6.34454139120765680077275474847, 7.04235951318806922424719211684, 7.57554585883996408500878316989, 8.803923550353660546531587111530