L(s) = 1 | + (−1.22 − 1.22i)3-s + (−5.44 + 5.44i)7-s + 2.99i·9-s + 6.44·11-s + (−14.4 − 14.4i)13-s + (23.1 − 23.1i)17-s + 16.6i·19-s + 13.3·21-s + (−6.65 − 6.65i)23-s + (3.67 − 3.67i)27-s + 0.0454i·29-s − 4.49·31-s + (−7.89 − 7.89i)33-s + (−35.3 + 35.3i)37-s + 35.3i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.778 + 0.778i)7-s + 0.333i·9-s + 0.586·11-s + (−1.11 − 1.11i)13-s + (1.36 − 1.36i)17-s + 0.878i·19-s + 0.635·21-s + (−0.289 − 0.289i)23-s + (0.136 − 0.136i)27-s + 0.00156i·29-s − 0.144·31-s + (−0.239 − 0.239i)33-s + (−0.955 + 0.955i)37-s + 0.907i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9134837520\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9134837520\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (5.44 - 5.44i)T - 49iT^{2} \) |
| 11 | \( 1 - 6.44T + 121T^{2} \) |
| 13 | \( 1 + (14.4 + 14.4i)T + 169iT^{2} \) |
| 17 | \( 1 + (-23.1 + 23.1i)T - 289iT^{2} \) |
| 19 | \( 1 - 16.6iT - 361T^{2} \) |
| 23 | \( 1 + (6.65 + 6.65i)T + 529iT^{2} \) |
| 29 | \( 1 - 0.0454iT - 841T^{2} \) |
| 31 | \( 1 + 4.49T + 961T^{2} \) |
| 37 | \( 1 + (35.3 - 35.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 20.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.2 - 32.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (50.5 - 50.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-5.50 - 5.50i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 55.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 47.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (85.2 - 85.2i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-21.9 - 21.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-94.9 - 94.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 71.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-37 + 37i)T - 9.40e3iT^{2} \) |
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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
−9.829752424318583896917928037292, −9.024916255594406796108727393402, −7.890847307584583181184697379434, −7.33819782001210578395206669562, −6.26705047671334290082017060210, −5.62965163750898561125636311123, −4.81615609996670574802233513871, −3.32135062171533998260440095972, −2.57274028360478869410010828879, −1.02687571808883546921744509434,
0.33663142202162034631448183360, 1.84710467920906373025609145796, 3.43134904005832387549738829898, 4.05552546070067765369016699479, 5.06699503737823611060079944460, 6.08887138132199645683454806145, 6.88116028216464584130233800025, 7.52462636110374345270550294838, 8.772746694425125187325223976670, 9.568082905246422101631492218790