L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (−0.0221 − 2.23i)5-s + (0.707 − 0.707i)6-s + (−0.281 − 0.281i)7-s + i·8-s + 1.00i·9-s + (−2.23 + 0.0221i)10-s − 1.64i·11-s + (−0.707 − 0.707i)12-s − 3.15i·13-s + (−0.281 + 0.281i)14-s + (1.56 − 1.59i)15-s + 16-s + 3.44·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.00990 − 0.999i)5-s + (0.288 − 0.288i)6-s + (−0.106 − 0.106i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.707 + 0.00700i)10-s − 0.496i·11-s + (−0.204 − 0.204i)12-s − 0.875i·13-s + (−0.0752 + 0.0752i)14-s + (0.404 − 0.412i)15-s + 0.250·16-s + 0.835·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.453094908\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.453094908\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.0221 + 2.23i)T \) |
| 37 | \( 1 + (6.01 - 0.925i)T \) |
good | 7 | \( 1 + (0.281 + 0.281i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.64iT - 11T^{2} \) |
| 13 | \( 1 + 3.15iT - 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 + (-1.85 - 1.85i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.91iT - 23T^{2} \) |
| 29 | \( 1 + (2.67 - 2.67i)T - 29iT^{2} \) |
| 31 | \( 1 + (6.77 + 6.77i)T + 31iT^{2} \) |
| 41 | \( 1 + 8.72iT - 41T^{2} \) |
| 43 | \( 1 + 1.27iT - 43T^{2} \) |
| 47 | \( 1 + (0.752 + 0.752i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.88 + 3.88i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.53 - 6.53i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.144 + 0.144i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.492 + 0.492i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + (0.804 + 0.804i)T + 73iT^{2} \) |
| 79 | \( 1 + (2.86 + 2.86i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.45 + 3.45i)T - 83iT^{2} \) |
| 89 | \( 1 + (12.5 - 12.5i)T - 89iT^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−9.628441824610487132149855653686, −8.762513968227471746814463910493, −8.236033898555009726887643095932, −7.31810416572421159102472745423, −5.65384298326321565491784272163, −5.24785125874958343780928248585, −3.98326946142540303008510784223, −3.37125549158279992889274226759, −2.02142117873643011054052773904, −0.61784406643691498223982933384,
1.70711366816363643101787780221, 3.03118668663650704533218169908, 3.90986612248543767134545444417, 5.19702673956171940730865997140, 6.14554085924327870379396838886, 7.05532851463685871701104748724, 7.37726410083307706485689111214, 8.332685481237964368577148469209, 9.378163176753709500244054600158, 9.815456848007737881798082303537