| L(s) = 1 | − 2i·7-s + 11-s − 4i·13-s + 6i·17-s − 8·19-s + 3i·23-s + 5·31-s + i·37-s − 10i·43-s + 3·49-s + 6i·53-s + 3·59-s − 4·61-s + i·67-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 0.301·11-s − 1.10i·13-s + 1.45i·17-s − 1.83·19-s + 0.625i·23-s + 0.898·31-s + 0.164i·37-s − 1.52i·43-s + 0.428·49-s + 0.824i·53-s + 0.390·59-s − 0.512·61-s + 0.122i·67-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6232763342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6232763342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−7.86673338407172572956157904367, −7.27040299268651647031701025867, −6.41303281959924649071685626748, −5.99997560843970877733772087164, −5.15827689266317656643755855851, −4.17855995155298485789427699803, −3.88466487321236647478728188393, −2.90909384127685555858687847921, −1.94680570161132183131972457098, −1.02894714600965483550008589065,
0.14465034863896903010185932707, 1.50570228281364337148201057885, 2.40986432391260054970842390184, 2.93107856349892987261432118937, 4.23954675601523496069781506958, 4.47986452872008914739451181266, 5.40751353344658329321732759029, 6.21935894487062543939440264722, 6.69155611585652152365917629733, 7.30443094222752784243650879642
