| L(s) = 1 | − 3-s + (−1 + 2i)5-s − 2i·7-s + 9-s − 2i·11-s + 6·13-s + (1 − 2i)15-s + 2i·17-s + 2i·21-s − 4i·23-s + (−3 − 4i)25-s − 27-s − 8·31-s + 2i·33-s + (4 + 2i)35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.447 + 0.894i)5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s + 1.66·13-s + (0.258 − 0.516i)15-s + 0.485i·17-s + 0.436i·21-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.192·27-s − 1.43·31-s + 0.348i·33-s + (0.676 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112749778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112749778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.325806178665803861551864609891, −7.46483876337736422533928003126, −6.84720043549624350358626348772, −6.14007842572282967512468526645, −5.60358478788117515773885525188, −4.25972264733339860129855668421, −3.82186515933564960511372617941, −2.99222608237856889134273527132, −1.60187568340872147871724595077, −0.42137465715308899645928942561,
1.03975617128410289720254036218, 1.93756372283623915499700052105, 3.35154896961385924792478773281, 4.11548539221463832232546862790, 4.97146855756344697124598267090, 5.62852528967901714807388011888, 6.22258883538268339401687359678, 7.23987004492467640026617087566, 7.908853449102611121118775583871, 8.734379806350938919729633373185
