L(s) = 1 | − 2.61i·2-s + 0.146i·3-s − 4.81·4-s + (0.948 + 2.02i)5-s + 0.383·6-s − 1.14i·7-s + 7.36i·8-s + 2.97·9-s + (5.28 − 2.47i)10-s − 5.36·11-s − 0.707i·12-s − 2.41i·13-s − 2.97·14-s + (−0.297 + 0.139i)15-s + 9.58·16-s + 5.74i·17-s + ⋯ |
L(s) = 1 | − 1.84i·2-s + 0.0847i·3-s − 2.40·4-s + (0.424 + 0.905i)5-s + 0.156·6-s − 0.431i·7-s + 2.60i·8-s + 0.992·9-s + (1.67 − 0.783i)10-s − 1.61·11-s − 0.204i·12-s − 0.669i·13-s − 0.796·14-s + (−0.0767 + 0.0359i)15-s + 2.39·16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.395835616\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.395835616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.948 - 2.02i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.61iT - 2T^{2} \) |
| 3 | \( 1 - 0.146iT - 3T^{2} \) |
| 7 | \( 1 + 1.14iT - 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 + 2.41iT - 13T^{2} \) |
| 17 | \( 1 - 5.74iT - 17T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 7.37iT - 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 - 7.59iT - 43T^{2} \) |
| 47 | \( 1 + 7.85iT - 47T^{2} \) |
| 53 | \( 1 - 0.0179iT - 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 0.0189T + 61T^{2} \) |
| 67 | \( 1 - 7.07iT - 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 - 5.58iT - 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 + 0.390iT - 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
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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
−9.819871305413674020720069813300, −8.441628889931931607718353396204, −7.889136228100834563586188865674, −6.75468202350052587570094786494, −5.60873442951207243110690202595, −4.67371486653452451622196355603, −3.83977054679522228433326834814, −2.92256559627493500534295115976, −2.25405736273278957126227221972, −1.05402843193078836445616293902,
0.64558946276575469602853696608, 2.41037768769957372606925248542, 4.17843227145394338050537996752, 4.91090690613221779738128957749, 5.37078061304481606352342647851, 6.22570608582442101456352885903, 7.13697938503611334474242891831, 7.70201258427677657717142284491, 8.428256399740846316471868875847, 9.221332538262764862706352071900