L(s) = 1 | − 0.288i·2-s − i·3-s + 1.91·4-s + (−0.407 − 2.19i)5-s − 0.288·6-s − 3.66i·7-s − 1.13i·8-s − 9-s + (−0.635 + 0.117i)10-s − 1.91i·12-s − 4.82i·13-s − 1.05·14-s + (−2.19 + 0.407i)15-s + 3.50·16-s + 3.84i·17-s + 0.288i·18-s + ⋯ |
L(s) = 1 | − 0.204i·2-s − 0.577i·3-s + 0.958·4-s + (−0.182 − 0.983i)5-s − 0.117·6-s − 1.38i·7-s − 0.399i·8-s − 0.333·9-s + (−0.200 + 0.0371i)10-s − 0.553i·12-s − 1.33i·13-s − 0.283·14-s + (−0.567 + 0.105i)15-s + 0.876·16-s + 0.931i·17-s + 0.0680i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.838384394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.838384394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.407 + 2.19i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.288iT - 2T^{2} \) |
| 7 | \( 1 + 3.66iT - 7T^{2} \) |
| 13 | \( 1 + 4.82iT - 13T^{2} \) |
| 17 | \( 1 - 3.84iT - 17T^{2} \) |
| 19 | \( 1 + 6.96T + 19T^{2} \) |
| 23 | \( 1 + 1.20iT - 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 - 2.38iT - 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 + 0.772iT - 43T^{2} \) |
| 47 | \( 1 - 6.47iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 + 6.10iT - 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + 0.191iT - 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 1.53iT - 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−8.496559582361561944840751924132, −8.143357621017417851540747056358, −7.37863628429082229787412669819, −6.52284822749005317256893832477, −5.89618559599660626410844082135, −4.65917297302881402801009404041, −3.84331159400100613282216571582, −2.73951317205864851328875345856, −1.49625172982917607367731472176, −0.65266803856422084632517866839,
2.18085903377251356348417843198, 2.56346869429824096849438455627, 3.68243320168229934425350249631, 4.81046539183561770667485570959, 5.78739076834306403250042564612, 6.59396362160799551735261861311, 6.92300129526018766064168900554, 8.207594787267630443741403139936, 8.725272038716219622256239906941, 9.739410773221860058115784895002