L(s) = 1 | + i·2-s + (1.68 − 0.420i)3-s − 4-s + (0.420 + 1.68i)6-s + (1.16 − 2.37i)7-s − i·8-s + (2.64 − 1.41i)9-s − 2.82i·11-s + (−1.68 + 0.420i)12-s + 0.841i·13-s + (2.37 + 1.16i)14-s + 16-s − 1.19·17-s + (1.41 + 2.64i)18-s + 4.55i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.970 − 0.242i)3-s − 0.5·4-s + (0.171 + 0.685i)6-s + (0.439 − 0.898i)7-s − 0.353i·8-s + (0.881 − 0.471i)9-s − 0.852i·11-s + (−0.485 + 0.121i)12-s + 0.233i·13-s + (0.635 + 0.311i)14-s + 0.250·16-s − 0.288·17-s + (0.333 + 0.623i)18-s + 1.04i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.228303088\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228303088\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.68 + 0.420i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.16 + 2.37i)T \) |
good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 0.841iT - 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 4.55iT - 19T^{2} \) |
| 23 | \( 1 + 3.29iT - 23T^{2} \) |
| 29 | \( 1 + 7.98iT - 29T^{2} \) |
| 31 | \( 1 + 5.53iT - 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 - 12.5iT - 53T^{2} \) |
| 59 | \( 1 + 3.91T + 59T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 - 4.65T + 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.06iT - 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 8.11iT - 97T^{2} \) |
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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
−9.706963130696336282715639793136, −8.853186546869516613146228222664, −7.987138579391957506962120468426, −7.68236822228043597606354948366, −6.60928982506391706704919941541, −5.84425615648474443592169376406, −4.36584231390007882062990898359, −3.89524228511810608492648182611, −2.51792064107248490324160511821, −0.989153278428979362414854182035,
1.63594264162412229358345708799, 2.53903713272494110986143488104, 3.43540531765408701190554538034, 4.65440299530035597327839077276, 5.20627364600485544027003704801, 6.73571852957405642929390376455, 7.69833020680841368745783410173, 8.542410998543180802779877461929, 9.192793716847264568032830745186, 9.765595169048570152343452642227