| L(s) = 1 | − 1.41·2-s + 3.79·3-s + 2.00·4-s − 5.36·6-s − 7.10i·7-s − 2.82·8-s + 5.39·9-s + 11.2i·11-s + 7.58·12-s − 20.0·13-s + 10.0i·14-s + 4.00·16-s − 1.63i·17-s − 7.62·18-s + 29.4i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.26·3-s + 0.500·4-s − 0.894·6-s − 1.01i·7-s − 0.353·8-s + 0.599·9-s + 1.02i·11-s + 0.632·12-s − 1.54·13-s + 0.717i·14-s + 0.250·16-s − 0.0959i·17-s − 0.423·18-s + 1.54i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3873099115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3873099115\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (20.0 + 11.3i)T \) |
| good | 3 | \( 1 - 3.79T + 9T^{2} \) |
| 7 | \( 1 + 7.10iT - 49T^{2} \) |
| 11 | \( 1 - 11.2iT - 121T^{2} \) |
| 13 | \( 1 + 20.0T + 169T^{2} \) |
| 17 | \( 1 + 1.63iT - 289T^{2} \) |
| 19 | \( 1 - 29.4iT - 361T^{2} \) |
| 29 | \( 1 + 50.3T + 841T^{2} \) |
| 31 | \( 1 - 11.1T + 961T^{2} \) |
| 37 | \( 1 + 40.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 7.24T + 1.68e3T^{2} \) |
| 43 | \( 1 - 71.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.40T + 2.20e3T^{2} \) |
| 53 | \( 1 - 20.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 65.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 124. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 43.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 48.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.63iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 143. iT - 9.40e3T^{2} \) |
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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
−9.871847578077245365067105037467, −9.261015263885493378566554617813, −8.199610088914261094334434881166, −7.50738779902735193436817547666, −7.26903848385712430708888309894, −5.89632468828465884459607119131, −4.49407583371607970267568736911, −3.67747707856465772641026303066, −2.48429761428066236859622816039, −1.68834486057218623450549244020,
0.10887474069670421116916412444, 2.02468783787197446006628229099, 2.66002136273633990409056298334, 3.53806725914053547607173384489, 5.03400710983096775198490785916, 5.96123456594637477023499422418, 7.12718603989557335413944420544, 7.84728654855416912138463792777, 8.581301422303127930279710424413, 9.214146440272946943591288578156
