L(s) = 1 | + 2-s + (1.57 + 0.721i)3-s + 4-s + (1.57 + 0.721i)6-s + (2.29 − 1.31i)7-s + 8-s + (1.95 + 2.27i)9-s + 3.91i·11-s + (1.57 + 0.721i)12-s − 4.99·13-s + (2.29 − 1.31i)14-s + 16-s + 3.54i·17-s + (1.95 + 2.27i)18-s − 3.14i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.909 + 0.416i)3-s + 0.5·4-s + (0.642 + 0.294i)6-s + (0.867 − 0.496i)7-s + 0.353·8-s + (0.652 + 0.757i)9-s + 1.18i·11-s + (0.454 + 0.208i)12-s − 1.38·13-s + (0.613 − 0.351i)14-s + 0.250·16-s + 0.860i·17-s + (0.461 + 0.535i)18-s − 0.722i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.599786633\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.599786633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.57 - 0.721i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.29 + 1.31i)T \) |
good | 11 | \( 1 - 3.91iT - 11T^{2} \) |
| 13 | \( 1 + 4.99T + 13T^{2} \) |
| 17 | \( 1 - 3.54iT - 17T^{2} \) |
| 19 | \( 1 + 3.14iT - 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 + 7.54iT - 29T^{2} \) |
| 31 | \( 1 - 4.19iT - 31T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 2.91iT - 43T^{2} \) |
| 47 | \( 1 + 8.00iT - 47T^{2} \) |
| 53 | \( 1 + 0.288T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 2.48iT - 61T^{2} \) |
| 67 | \( 1 - 0.545iT - 67T^{2} \) |
| 71 | \( 1 - 5.37iT - 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 0.742T + 79T^{2} \) |
| 83 | \( 1 + 4.45iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.524T + 97T^{2} \) |
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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
−10.06633061171544159899091028623, −9.214503379631296788646429053900, −8.229113580418801753059488555446, −7.34656945460549348019265033976, −6.95252730525519930050833791991, −5.21698851324653698712306666191, −4.68922152510749766653841881123, −3.93068172163531910796164885847, −2.65826475882876974152882287679, −1.79706509620442974308830948049,
1.39945618027725773384606748354, 2.67125908763428994749090689218, 3.29063580519127937575970038627, 4.69497284664499641867530557594, 5.33061330869725299085076577720, 6.54395833369080102687373336356, 7.36167418722789978101541170061, 8.119963738231020210090555412707, 8.873651215762974944864539359371, 9.711542937415637899145048053443