| L(s) = 1 | + 1.47·2-s + 0.172·4-s + 4.57·7-s − 2.69·8-s − 5.57·11-s + 0.102·13-s + 6.74·14-s − 4.31·16-s + 5.83·17-s + 2.31·19-s − 8.22·22-s − 6.14·23-s + 0.150·26-s + 0.790·28-s + 3.74·29-s + 8.49·31-s − 0.975·32-s + 8.59·34-s + 1.46·37-s + 3.41·38-s + 2.75·41-s + 2.49·43-s − 0.964·44-s − 9.05·46-s − 8.61·47-s + 13.9·49-s + 0.0176·52-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0864·4-s + 1.72·7-s − 0.952·8-s − 1.68·11-s + 0.0283·13-s + 1.80·14-s − 1.07·16-s + 1.41·17-s + 0.531·19-s − 1.75·22-s − 1.28·23-s + 0.0295·26-s + 0.149·28-s + 0.694·29-s + 1.52·31-s − 0.172·32-s + 1.47·34-s + 0.241·37-s + 0.553·38-s + 0.430·41-s + 0.380·43-s − 0.145·44-s − 1.33·46-s − 1.25·47-s + 1.98·49-s + 0.00245·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.417071249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417071249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 0.102T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 - 2.49T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + 6.37T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 8.92T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−8.054689972243998394518048795713, −7.68531423610929101708421474455, −6.47422973095316571175573586455, −5.59472535718465203414767354384, −5.15655340704435687992086826635, −4.69921377665217540778013636849, −3.83780793201766047111517297413, −2.89809240286931647638286225111, −2.15278949354611503073444771186, −0.854963949968328608349272102671,
0.854963949968328608349272102671, 2.15278949354611503073444771186, 2.89809240286931647638286225111, 3.83780793201766047111517297413, 4.69921377665217540778013636849, 5.15655340704435687992086826635, 5.59472535718465203414767354384, 6.47422973095316571175573586455, 7.68531423610929101708421474455, 8.054689972243998394518048795713
