| L(s) = 1 | − 2i·3-s − 2i·7-s − 9-s + 2i·13-s + 6i·17-s + 4·19-s − 4·21-s + 6i·23-s − 4i·27-s − 6·29-s − 4·31-s − 2i·37-s + 4·39-s + 6·41-s − 10i·43-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.755i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s − 0.872·21-s + 1.25i·23-s − 0.769i·27-s − 1.11·29-s − 0.718·31-s − 0.328i·37-s + 0.640·39-s + 0.937·41-s − 1.52i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.865143 - 0.534688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.865143 - 0.534688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−13.52643273303449432817387322879, −12.81002754851319816436699890857, −11.75083487116390041285185985182, −10.62485914808997747602449648235, −9.284859191225383023872534913133, −7.79001924966433881829146023073, −7.11279555676085094514050237941, −5.81551045176731723788667961676, −3.87154079940685422623428227102, −1.62617164730213689031770455141,
3.00312250730576770700466962261, 4.61840726780746961176124773780, 5.68482641200058823791632445172, 7.43493432744747467519834720266, 8.960199626410135445348724291747, 9.678494221603797481097103622902, 10.78630433012170936477774500537, 11.81746361245409757385621519176, 12.98781540167559015103629201578, 14.33024189415308501730553576776
