L(s) = 1 | + 2.56i·2-s − 1.56i·3-s − 4.56·4-s + 4·6-s − 6.56i·8-s + 0.561·9-s − 1.56·11-s + 7.12i·12-s − 0.438i·13-s + 7.68·16-s − 0.438i·17-s + 1.43i·18-s − 7.12·19-s − 4i·22-s + 3.12i·23-s − 10.2·24-s + ⋯ |
L(s) = 1 | + 1.81i·2-s − 0.901i·3-s − 2.28·4-s + 1.63·6-s − 2.31i·8-s + 0.187·9-s − 0.470·11-s + 2.05i·12-s − 0.121i·13-s + 1.92·16-s − 0.106i·17-s + 0.339i·18-s − 1.63·19-s − 0.852i·22-s + 0.651i·23-s − 2.09·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3792619192\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3792619192\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.56iT - 2T^{2} \) |
| 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438iT - 13T^{2} \) |
| 17 | \( 1 + 0.438iT - 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876iT - 43T^{2} \) |
| 47 | \( 1 + 8.68iT - 47T^{2} \) |
| 53 | \( 1 + 5.12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 5.80iT - 97T^{2} \) |
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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
−9.177456625756241974464176340974, −8.379721186546144565767277298670, −7.72039885121217685018724221753, −7.12271466091145166551235524173, −6.41902946282955143987110423168, −5.69265867587008291929648008770, −4.77744983512535875085703793078, −3.77930030276753349828525289941, −1.97136371300973585458459343513, −0.15957125003530968475800879426,
1.60962828694916059929954737449, 2.67720231338740483210153844531, 3.68620816984602138141416592199, 4.40994731801527051968359674453, 5.02734117768853056709916644152, 6.33564363000852839777055050728, 7.77502262897488909931505251715, 8.799552840339272578681098390358, 9.304591871833970065093728019476, 10.20023478818217549162953008509