L(s) = 1 | + 1.61·2-s − 3-s + 0.597·4-s + 4.27·5-s − 1.61·6-s + 1.98·7-s − 2.25·8-s + 9-s + 6.89·10-s + 11-s − 0.597·12-s + 0.250·13-s + 3.20·14-s − 4.27·15-s − 4.83·16-s + 3.17·17-s + 1.61·18-s − 3.76·19-s + 2.55·20-s − 1.98·21-s + 1.61·22-s + 8.19·23-s + 2.25·24-s + 13.3·25-s + 0.404·26-s − 27-s + 1.18·28-s + ⋯ |
L(s) = 1 | + 1.13·2-s − 0.577·3-s + 0.298·4-s + 1.91·5-s − 0.658·6-s + 0.751·7-s − 0.798·8-s + 0.333·9-s + 2.18·10-s + 0.301·11-s − 0.172·12-s + 0.0695·13-s + 0.856·14-s − 1.10·15-s − 1.20·16-s + 0.769·17-s + 0.379·18-s − 0.863·19-s + 0.572·20-s − 0.433·21-s + 0.343·22-s + 1.70·23-s + 0.461·24-s + 2.66·25-s + 0.0792·26-s − 0.192·27-s + 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.788116824\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.788116824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 4.27T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 13 | \( 1 - 0.250T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 - 0.852T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 - 9.77T + 59T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 2.07T + 71T^{2} \) |
| 73 | \( 1 + 1.20T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−9.186610930281698015579817370905, −8.620722289465538675293143708421, −7.17295189082387233857344486496, −6.38112711559247772533187782231, −5.79423895609877617742206660194, −5.13334687574432185745942462931, −4.65771167804981718879506660645, −3.36978610695397873972395898322, −2.29138802434919553474741042215, −1.28075029946974639043645695509,
1.28075029946974639043645695509, 2.29138802434919553474741042215, 3.36978610695397873972395898322, 4.65771167804981718879506660645, 5.13334687574432185745942462931, 5.79423895609877617742206660194, 6.38112711559247772533187782231, 7.17295189082387233857344486496, 8.620722289465538675293143708421, 9.186610930281698015579817370905