| L(s) = 1 | − i·2-s − 3-s − 4-s + 2i·5-s + i·6-s − 4i·7-s + i·8-s + 9-s + 2·10-s + 4i·11-s + 12-s − 4·14-s − 2i·15-s + 16-s − 2·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.894i·5-s + 0.408i·6-s − 1.51i·7-s + 0.353i·8-s + 0.333·9-s + 0.632·10-s + 1.20i·11-s + 0.288·12-s − 1.06·14-s − 0.516i·15-s + 0.250·16-s − 0.485·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9362556469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9362556469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.984627274575060375154472001668, −9.186426532963130803432971422905, −7.82777845582249317914993977072, −6.92354553957559178701022156738, −6.64517739130778337224491331814, −4.93249340136796759640583437805, −4.39671533942369886798902617987, −3.29442681378390684011724918485, −2.07276485331727122376890374480, −0.50529593574935926507859947205,
1.32191087055635942460036644758, 3.03709658692992629011188279084, 4.42655139886624894493651389595, 5.32159253812186055685560300117, 5.90340409199760937038378803657, 6.53178507873733559907668915509, 8.039406338575896882994620349769, 8.475884192090031632838804482682, 9.159884581276771846693277999461, 10.06757094813987388621180501153
