| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.23i·5-s − 8.87i·7-s + 2.82·8-s + 3.16i·10-s − 6.26i·11-s + 22.5·13-s − 12.5i·14-s + 4.00·16-s − 9.43i·17-s − 10.5i·19-s + 4.47i·20-s − 8.86i·22-s + (−22.7 + 3.26i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.447i·5-s − 1.26i·7-s + 0.353·8-s + 0.316i·10-s − 0.569i·11-s + 1.73·13-s − 0.896i·14-s + 0.250·16-s − 0.555i·17-s − 0.554i·19-s + 0.223i·20-s − 0.403i·22-s + (−0.989 + 0.141i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.140385522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140385522\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (22.7 - 3.26i)T \) |
| good | 7 | \( 1 + 8.87iT - 49T^{2} \) |
| 11 | \( 1 + 6.26iT - 121T^{2} \) |
| 13 | \( 1 - 22.5T + 169T^{2} \) |
| 17 | \( 1 + 9.43iT - 289T^{2} \) |
| 19 | \( 1 + 10.5iT - 361T^{2} \) |
| 29 | \( 1 - 7.04T + 841T^{2} \) |
| 31 | \( 1 + 18.9T + 961T^{2} \) |
| 37 | \( 1 - 39.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 2.96T + 1.68e3T^{2} \) |
| 43 | \( 1 + 73.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 64.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 14.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 19.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 55.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 41.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 33.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 40.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 49.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 36.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−8.588358645351726293028441633199, −7.892098806370418041249584749245, −6.97649121586928943947317600550, −6.44370716632000163112545301889, −5.62507369330588071806624559113, −4.57276166822369133452520959165, −3.68997108203352661777480408835, −3.24016841346384668312777994567, −1.79083914266881238681052192775, −0.60655385013868643646225114882,
1.40894693411189204778865968164, 2.24969681837166923573207556523, 3.44872420723315517074808038494, 4.19879509465711034109341830192, 5.17962841048287228158981379584, 6.02450900319830743002121369556, 6.28791564479650119132685390595, 7.64711081511481603404896941872, 8.373755361980024149061560766430, 8.975271182540939278793669859191
