| L(s) = 1 | + i·2-s − i·3-s − 4-s + (2 + i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (−1 + 2i)10-s + 2·11-s + i·12-s + i·13-s − 2·14-s + (1 − 2i)15-s + 16-s + 2i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s + 0.603·11-s + 0.288i·12-s + 0.277i·13-s − 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28423 + 0.793703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28423 + 0.793703i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−11.67929741531403913953414891160, −10.40349342832384002536881095366, −9.428101240143841676202585748156, −8.720742915198779871262138883141, −7.61178114907466501403572635929, −6.57053871516389154097696472168, −6.00873995796246908305735998507, −4.96935059804846759672691954704, −3.21911290967116870543334881841, −1.74012033866586859306908382209,
1.18946337852103836515029025778, 2.85808539939422729727876459943, 4.13125857353407705589556702147, 5.07192883066494602073126526013, 6.14513308344513577840423000173, 7.52845205699816026568328387423, 8.794398656612434567960187662607, 9.559245089767598976866079494335, 10.15182428810637106566355532472, 11.03988785741767423611475136287
