| L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s + 0.374·7-s − i·8-s + 9-s + 10-s − 6.11·11-s − 12-s − 5.44i·13-s + 0.374i·14-s − i·15-s + 16-s − 2.37i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.141·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.84·11-s − 0.288·12-s − 1.51i·13-s + 0.0999i·14-s − 0.258i·15-s + 0.250·16-s − 0.575i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.235535148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235535148\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (-2.70 - 5.44i)T \) |
| good | 7 | \( 1 - 0.374T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 + 5.44iT - 13T^{2} \) |
| 17 | \( 1 + 2.37iT - 17T^{2} \) |
| 19 | \( 1 + 1.70iT - 19T^{2} \) |
| 23 | \( 1 + 5.44iT - 23T^{2} \) |
| 29 | \( 1 - 4.41iT - 29T^{2} \) |
| 31 | \( 1 + 3.07iT - 31T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 7.82iT - 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 4.07iT - 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 6.78T + 83T^{2} \) |
| 89 | \( 1 + 0.551iT - 89T^{2} \) |
| 97 | \( 1 + 8.15iT - 97T^{2} \) |
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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
−9.643250500136900503648216274674, −8.527023911218116531486414470722, −8.077914705569534795624242974827, −7.48494037481886461616157557339, −6.36763560389966248533193120623, −5.13990249154716525426299727930, −4.93264080339173162798563934522, −3.37390782374923872688219762636, −2.45048548525437259149616380828, −0.48672693371614329752513174304,
1.75279153798382622683326816128, 2.61108377119400778804633141053, 3.61934492578378904858909251691, 4.57774513581011626737657588749, 5.59254541670223229303590087282, 6.74522537139669652952774237046, 7.82058160573652209554251003569, 8.248093614921785663155001568053, 9.472237342983389740511704198304, 9.863274838488575138934152885464
